Extended Pythagoras Theorem using Triangles, and its Applications to Engineering
Engineering is governed by mathematical laws, of which a pearl is the Pythagoras Theorem x2+y2=z2 - an equation that has enabled humans to solve problems for the past millennia in all fields of science, ranging from civil to electrical engineering. In view of unlocking further benefits, the present article looks at a variant equation of the Pythagoras Theorem, namely x2+xy+y2=z2 . Here, it was found that just as the Pythagoras Theorem x2+y2=z2 establishes an area relation between squares along the sides of a right-angled triangle, equation x2+xy+y2=z2 extends this theorem by establishing an area relation between triangles along the sides of a 120-degree-angled triangle. Proofs are presented in both cases - including geometric and algebraic components – and are followed by classroom exercises, showing that both classical and extended versions of the Pythagoras Theorem are equally useful in cases of mathematics applied to engineering.
1.1 Engineering and Orthogonality
We are surrounded by orthogonality, very often triggered by gravity. The simplest expression of it is a vertical wall standing on a horizontal ground. Keeping it vertical sounds like a trivial problem, but preventing vertical structures from falling has been a long lasting endeavor of humankind that is still a challenge today. Transferring information (like for instance, loads) from a vertical direction to a horizontal – which are by definition independent or decoupled - implies a diagonal path. For example, in civil engineering bridges connect an horizontal platform at length x to a vertical column of height y via intermediate diagonal beams of length z (Figure 1a).
Similarly, in aerospace engineering a flying airplane with a (ground) forward velocity vector x encountering a side wind with an orthogonal velocity vector y, together give a diagonal true air velocity vector z, towards which the aircraft is flying relative to the air (Figure 1b). Knowing the true air behavior around an aircraft is critical to allow the aerodynamic surfaces to be controlled in the proper way to guide the aeroplane to fly in the desired direction. The interrelation between these three lengths, governed by orthogonality, is by definition the very essence of the Pythagoras Theorem
and, these examples show just how important this theorem is in solving problems in engineering. Since the Pythagoras Theorem emerged from the existence of orthogonal situations that are present in the world, would it stand to reason to think that each of the remainder angles has a theorem of its own under equivalent situations? What would they look like? This article will explore one such variant of the theorem, where the reference angle is not 90 degrees (orthogonal), but 120 degrees instead.
1.2 Babylonians and the cane against the wall
Practical problems involving the application of the Pythagoras Theorem dates back to the times of the Babylonians, where measuring the length of a cane leaning against a wall was a real challenge (Figure 2a) . For the case where the wall is orthogonal to the ground (forming a right-angled triangle), the Pythagoras Theorem has proven to be the right tool to answer this riddle. But what happens to the theorem when the wall or the floor is inclined, and the new angle between them changes to 120 degrees?Let us start by assuming the simple case where the length of the cane along the ground to the wall is 3 meters, i.e., 𝑥=3, and the length of the cane along the wall from the ground is 4 meters, i.e., 𝑦=4. The Pythagoras Theorem gives us the direct answer 32+42=52, resulting in a cane length of 5 meters, i.e., 𝑧=5. Now imagine that the ground is in fact inclined, and the actual angle between the wall and the ground is 120 degrees (Figure 2b). Since this is no longer a right-angled triangle, the Pythagoras Theorem can no longer be applied as the necessary premise of orthogonality is no longer present.
In this new situation, we must find another equation to determine the new length of the cane. Now, one may be tempted to use the available trigonometric function of the law of cosines. According to Euclid , the relation between the area of the three squares surrounding an obtuse triangle (Figure 3) is expressed as
While 𝐴𝐵=𝑧, 𝐶𝐴=𝑥 and 𝐶𝐵=𝑦 are known, evaluation of length CH was unknown. Thus, Euclid's Elements (300 B.C.) provided the foundation that allowed for the law of cosines, with its modern expression (evolving from trigonometric tables towards its final theorem) was only presented later in the fifteenth century . Replacing 𝐶𝐻=𝑦𝑐𝑜𝑠(𝜋−𝛾)=−𝑦𝑐𝑜𝑠(𝛾) in Eq.(2) gives
where 𝛾 is the largest internal angle, which is the law of cosines. However, this cannot be seen as a true extension of the Pythagoras theorem as it lacks the sum-of-areas geometrical proof. By expanding cos(γ) as a Taylor series, it is easy to see that any attempt to express it geometrically as a sum of areas is, at least, a daunting task.
This is not really what is intended in an equation that expresses the extended application of the Pythagoras Theorem to relate triangles instead of squares. Triangles because just as a square has an external angle of 90 degrees and governs the Pythagoras Theorem, a triangle has an external angle of 120 degrees and here is hypothesized to govern the new extended version of the theorem (this will be verified later in chapter 3). Corroborating this point from another perspective, just as lines that start at the corners of a square and meet at the center divide the square into four isosceles right-angled triangles (i.e., a particular case of the triangle in Figure 2a), similarly lines starting at the corners of an equilateral triangle that meet at the center divide the triangle into three isosceles 120-degree triangles (i.e., a particular case of the triangle in Figure 2b). Placing ourselves in the shoes of Euclid, Pythagoras, and even the Babylonians, not knowing anything about cosine functions, they would probably try to find a solution that only uses what is available at the time – that is, the lengths of the side of the 120-degree-angled triangle in question. From this approach, our ideal function will be as simple as the original Pythagoras Theorem in its application, requiring only the knowledge of sides 𝑥 and 𝑦 to find the solution to the unknown diagonal side 𝑧.
Finally, the purpose of this article is to deliver that formula — while presenting its geometric and algebraic proof – that acts as the extended version of the Pythagoras Theorem governing the implied relations within a 120 degree-angled triangle. Indeed this is possible, as a precursor work — the publishing in 2021 of the proof of the variant of the Pythagoras Theorem using hexagons — forms a foundation, related in a parallel and complementary manner, to the proof to be presented in this article .
2 Pythagoras Theorem Revisited
The answer to the new riddle in Figure 2b is found by first revisiting the classical square-binding Pythagoras Theorem from the perspective of the central square theory .
Theorem 1 (Classical Pythagoras Theorem using Squares). If x, y and z are real numbers, then the lengths of the sides 𝑥 and 𝑦 of a right-angled triangle - forming the right angle between themselves - are related to the hypotenuse of the triangle z via the sum of area of the squares annexed to each corresponding side, such that
Proof. Many proofs of the Pythagoras Theorem have been presented over the years . One proof of the Pythagoras Theorem is based on the aforementioned central square approach [5,7] and has the geometrical expression shown in Figure 4. It is presently shown that for the general case 𝑥≠𝑦, the split of the combined area of squares 𝑥2 and 𝑦2 is shown to fit completely in the square of 𝑧2. This is now further explained. From Figure 4a, the square EAOF associated with the side 𝑦 can be decomposed into the central square KLOB’, the square EPKQ side 𝑥, plus the two side rectangles PALK and QKB’F of area 𝑥(𝑦−��) each.
One small rectangle PALK plus the 𝑥2 square EPKQ, form together a larger rectangle EALQ. Another larger rectangle is QKH’G’ is formed by displacing the 𝑥2 square OBHG to FB’H’G’. Both rectangles EALQ and QKH’G’ and the central square KLOB’ compose the total area of the sum 𝑥2 and 𝑦2. These three elements – EALQ, QKH’G’ and KLOB’ – are transferred to Figure 4b. In Figure 4b, splitting these rectangles along their diagonal sides QA and QH’ gives four right-angled triangles. The triangle QEA displaces to fill the space in AOD. Similarly, the triangle QG’H’ is displaced to fill the space DB’H’. Concluding, all the area of the smaller square OBHG in Figure 4a of area 𝑥2 and EAOF of area 𝑦2 is accounted for when forming the hypotenuse square ADH’Q of area 𝑧2 in Figure 4b .
The associated algebraic process of adding these areas, leading to equation 𝑥2+𝑦2=𝑧2, is described as follows. The left-hand side of the intended equation shows the larger square EAOF in Figure 4a of area 𝑦2 to be formed by a smaller square EPKQ of area 𝑥2 added to the central square KLOB’ of area (𝑦−𝑥)2 plus two rectangles PALK and QKB’F of combined area 2𝑥(𝑦−𝑥), resulting in
Likewise in Figure 4b, the right-hand side of the intended equation is formed by realizing that the square ADH’Q of area 𝑧2 is formed by four congruent right-angled triangles – AOD, DB’H’, H’KQ and QLA - of area 2xy each around a central square KLOB’ of area (𝑦−𝑥)2 which gives
Subtracting Eq.(4) from Eq.(5) removes the linking term – the central square (𝑦−𝑥)2 – and Eq.(5) becomes
Expanding the terms on the left, and switching sides for 𝑦2 results in
This concludes in the required Eq.(1) as
The proof is complete.
2.1 Particular case x=y
The particular case of 𝑥=𝑦 is a good starting point to explore the transition between the classical Pythagoras theorem to its extended triangular version (discussed in section 3.1). Here, we begin by presenting the geometrical proof based on the central square approach . This will enable a read across into the extended version, making it easier to understand the transition into the new version. Start with the square FABG inscribed in a circle (Figure 5). For the particular case 𝑥=𝑦, the right-angled triangle is formed by drawing lines from points A and B to the center of the square FABG at point M (i.e., along the x- and 𝑦-axis), which is here also coicident with the center of axes O. By extending those lines MA and MB to the other corners of the square - points F and G - results in the splitting of the square FABG into four isosceles right-angled triangles (i.e., AMB, BMG, GMF and FMA).
The aforementioned four triangles are placed on the outside (to the right), along the edge AB, forming the outer square ACDB with area 𝑧2 (placing this on the right of segment AB is inline with the proof in Figure 4). The square associated with the side 𝑦=MA (in this case equal to 𝑥) is obtained by drawing a line perpendicular to MF giving line FE, and perpendicular to MA giving EA, resulting in square FEAM of area 𝑥2. Similarly, the same can be done to the side 𝑥=MB resulting in the other square MBHG also with area 𝑥2. These geometrical results can be achieved with any available CAD software, of which open-source examples are the programs FreeCAD  and Geogebra .
It is seen in Figure 5 that the four right-angled triangles – FEA, FMA, GMB and GHB – forming the two smaller squares FEAM and GMBH with a combined area of 𝑥2+𝑥2, are found inside the square ACDB of area 𝑧2 – as AM’C, CM’D, DM’B and BM’A - fulfilling the end relation
Figure 5 is the foundation from which the particular case of the triangle version of the extended Pythagoras Theorem will be derived.
2.2 General case y=x
Moving now to the general case 𝑥≠𝑦, using as foundation the former particular case 𝑥=𝑦, imagine the larger square ACDB in Figure 5 of area 𝑧2 remains the same in size, but now rotates around the circle, as in Figure 6. The inner square FABG inscribed in the circle also rotates to a new position. The former isosceles right-angled triangle AMB in Figure 5 changes as side 𝑥 is no longer the same as 𝑦, and both move away from the orthogonal axes. The new sides 𝑥 and 𝑦 of the triangle can be found as follows. Draw an inward line from point A parallel to the 𝑦-axis, while at the same time draw another inward line from point B parallel to the 𝑥-axis. When both lines meet at point M gives the new scalene right-angled triangle AMB (in grey). Repeating this exercise starting at all other three corners of the square - i.e., points B, G and F – gives an additional three right angle triangles (BLG, GKF and FNA), all congruent. Together they shape the central square LKNM.
Following the layout of the proof presented earlier in Figure 4, these revolving triangles are mirrored outside of the circle (to the right, including the new central square N’K’L’M’) within the outer square ACDB of area 𝑧2. The square associated with the larger side 𝑦=MA is obtained by drawing a line starting at M and perpendicular to MA, all the way to the circle at point I, making the segment MI. Enclosing with the successively perpendicular segments IE and EA gives the square IEAM with area 𝑦2. Similarly, the square associated with the smaller side 𝑥=MB is obtained by extending M perpendicular to MB all the way to point J at the circle forming line MJ. Enclosing with successively perpendicular segments JH’ and H’B gives the square JMBH’ with area 𝑥2.
Algebraically, the squares JMBH’ and IEAM give a combined area of 𝑥2+𝑦2 that has been shown in Proof 1 to be equal to the area 𝑧2 of square ACDB, thus concluding in the relation
Figure 6 is the foundation from which the general case 𝑥≠𝑦 of the triangle version of the Pythagoras Theorem will be derived.
3 Pythagoras Theorem extended using Triangles
While the classical Pythagoras theorem operates in an environment where the lengths of the sides of the right-angled triangle are measured against an orthogonal (x- and y-axis) system, in the extended theorem the environment changes such that the lengths of the sides of the 120-degree triangle are measured against a triple (x-, y- and w-axis) system with an angular space of 60 degrees between each other.
3.1 Particular case x=y
Assume the length of the smaller sides 𝑥 and 𝑦 of the 120-degree-angled triangle are the same. Also, for this particular case 𝑥=𝑦, assume the prior square FABG in Figure 5 is now replaced by an equilateral triangle DAB in Figure 7 (both inscribed in the same circle). The triangle establishing the area relation changes from a right-angled triangle to an isosceles triangle with an obtuse angle of 120 degrees. This 120-degree-angled triangle is formed in the same manner as explained in section 2.1, that is by drawing lines from points A and B to the center of the equilateral triangle DAB at point M, which is coincident with the origin point O of the triple axes. Extending an additional line parallel to the w-axis from point M to the other corner of the triangle at point D results in the splitting of the equilateral triangle DAB into three 120-degree-angled triangles – AMB, BMD and DMA. Being consistent with the Classical Pythagoras Theorem presented earlier (i.e., Figure 4-6), the aforementioned three triangles are placed to the right on the outside of the circle, along the edge AB – becoming AO’C, CO’B and BO’A – all together forming the outer equilateral triangle ACB of area 43z2.
The equilateral triangle EMA of area 43x2 associated with the side 𝑦=MA is obtained by drawing a line upwards starting at point M, parallel to the x-axis, all the way to the circle at point E. Enclosing it with line EA gives the desired triangle EMA. The other equilateral triangle FMB, also of area 43x2, is obtained by drawing a line downwards starting at point M parallel to the y-axis, all the way to the circle at point F, and then enclosing it with line FB. It is observable that the combined area of these two equilateral triangles EMA and FOB together do not form the area of the outer larger equilateral triangle ACB of area 43z2. In fact, the combined area of the triangles EMA and FMB is 43(x2+x2), which can be decomposed into four right-angled triangles (EGO, AGM, FIM and BIM), while the outer larger triangle ACB is composed of six right-angled triangles (AG’O’, CG’O’, CI’O’, BI’O’, BH’O’ and AH’O’). By careful comparison, it becomes apparent that the missing additional two right-angled triangles (inside the circle) are EHM and FHM, representing a combined coupling triangle EMF of area 43x2, that links the former two equilateral triangles EMA and FMB of area 43x2 each.
Then it is straightforward to see that the sum of the areas 43(x2+x2) of the equilateral triangles EMA and FMB plus the coupling area 43x2 of the linking triangle EMF (all together, the three form the total area enclosed by EAMBF) gives the six right-angled triangles that form the equilateral triangle ACB of area 43z2. Thus, from an algebraic point of view, the area balance for this particular case 𝑥=𝑦 of the extended (triangle-based) version of the Pythagoras Theorem becomes
Where the coefficient 43 can be conveniently cancelled on either side, resulting in
3.2 General case y=x
Imagine the outer equilateral triangle ACB of area 43z2 in Figure 7 remains the same size, but now rotates clockwise around the circle to the position shown in Figure 8. The inner mirrored equilateral triangle DAB also of area 43z2 (inside the circle) displaces along its edges, making the reference 120-degree-angled triangle change size, shape and position. For the general case 𝑥≠𝑦, the now scalene 120-degree-angled triangle AMB in Figure 8, with the length of its sides 𝑥 and 𝑦 becoming different from each other, moves away from the corresponding axes. These new sides of the 120-degree-angled triangle can be found as before. Draw a line from point A downwards parallel to the 𝑦-axis, while at the same time draw a line upwards from point B parallel to the 𝑥-axis. When both lines meet at point L forms the new 120-degree-angled triangle AMB (filled in grey). Repeating this exercise starting at the other two corners of the equilateral triangle DAB - i.e., points B and D – gives an additional two scalene 120-degree-angled triangles BKD and DLA that (with AMB) are congruent. That is, imagine a line emerging simultaneously from each of the three corners of the equilateral triangle DAB, in an inward direction and parallel to each of the x-, y- and w-axis. These lines will eventually cross each other at points K, L and M - forming the segments AM, BK and CL. Since these three lines are parallel to each axis, they form an angle of 60 degrees between themselves. Since they do not meet in a single point (as they did in point M≡O in Figure8), three lines forming 60 degrees amongst each other will inherently form a triangle. Due to symmetry between all three axis, all the three lines AM, BK and DL are the same, but symmetrically rotated by 120 degrees around point O, which means they cross each other at the same location, pointing to an equal distance between the ends of the segments (for example, point M for segment AM) and the end of the segment (in this example, point A). Hence, segments LM, MK and KL have the same length, form internally 60 degrees amongst each other (or 120 degrees externally), and compose the sides of a triangle. The end conclusion is that these segments, all together under the defined constraints, form amongst themselves (internally to DAB) a central equilateral triangle KML.
Following the prior approach from Figure 5-7, the three revolving triangles - AMB, BKD and DLA - are mirrored outside the circle (to the right) - as AL’C, CK’B and BM’A - within the outer equilateral triangle ACB of area 43z2. Together, they form internally the mirrored central triangle M’K’L’. The equilateral triangle EMA associated with the larger side 𝑦=MA is obtained by drawing a line upwards starting at point M and inline with the 𝑥-axis all the way to the circle at point E to form segment ME. Enclosing with line EA gives the equilateral triangle EMA of area 43y2. Similarly, the equilateral triangle FMB associated with the smaller side 𝑥=M𝐵 is obtained by extending from point M downwards inline with the 𝑦-axis all the way to the circle at point F, forming the segment MF. Enclosing with line FB gives the equilateral triangle FMB of area 43x2. The coupling triangle EMF is obtained by connecting point E to F, giving a 120-degree-angled triangle of area 43xy, that links together the aforementioned two equilateral triangles EMA of area 43y2 and FMB of area 43x2.
Theorem 2 (Extended Pythagoras Theorem using Triangles). If x, y and z are real numbers, then the lengths of sides x and y of a 120-degree-angled triangle - forming the 120-degree angle between themselves - are related to the length of the longest side of the triangle z via the sum of the areas of the equilateral triangles that are annexed to each corresponding side, plus the area of a coupling 120-degree-angled triangle, resulting in
Proof. The geometrical expression of the extended version of the Pythagoras Theorem using triangles is shown in Figure 9, which emerges as an intentional evolution from the Classical Pythagoras Theorem shown previously in Figure 4. For the general case of 𝑥≠𝑦, the splitting of the area composed of the smaller equilateral triangles 43x2 and 43y2 plus the coupling 120-degree-angled triangle (Figure 9a) is shown to fit the totality of the area of the equilateral triangle 43z2 (Figure 9b). This is now further explained.
Figure 9a shows the reference 120-degree-angled triangle AB’B that replaces here the traditional right-angled triangle. At the bottom, along the side 𝑥=B’B of the 120-degree-angled triangle AB’B, there is the equilateral triangle FB’B of area 43x2, which replaces the traditional square 𝑥2 found in the classical theorem. At the top of Figure 9a, along the side 𝑦=AB’, there is the equilateral triangle EB’A of area 43y2 that can be decomposed into the central triangle D’A’B’ of area 43(y−x)2, another equilateral triangle HJI associated with side 𝑥=HJ of area 43x2, plus two side skewed rectangles EHJD’ and IAA’J. These two equilateral triangles EB’A of area 43y2 and FB’B of area 43x2 are coupled by a 120-degree-angled triangle EB’F of area 43xy. To the right of the reference 120-degree-angled triangle AB’B, there is the equilateral triangle ACB associated with side 𝑧=AC of area 43z2.
This is repeated and dealt separately in Figure 9b, being composed of three congruent 120-degree-angled triangles – AA’C, CD’B and BB’A – plus the central triangle A’D’B’. In Figure 9a, the equilateral triangle FB’B of area 43x2 (BB’F in Figure 9a), the two small skewed rectangles EHJD’ and IAA’J (LKMF and B’D’KL in Figure 9b) plus the equilateral triangle HJI of area 43x2 (D’CM in Figure 9b) form together in Figure 9b a larger skewed rectangle BD’CF of sides 𝑥=BF and 𝑦=FC. Splitting the larger skewed rectangle BD’CF (in Figure 9b) along its diagonal BC gives two congruent 120-degree-angled triangles. One fills the space in BD’C and the other BFC is displaced to fill the space in AA’C. The 120-degree-angled triangle in Figure 9a EB’F becomes AB’B in Figure 9b. The area of a 120-degree-angled triangle (like in Figure 9a AB’B or in Figure 9b AA’C) is found in the following manner. It is seen at the bottom of Figure 9b that the larger skewed rectangle BD’CF of sides 𝑥=BF and 𝑦=FC is formed by two smaller equilateral triangles BB’F and D’CM and two smaller skewed rectangles LKMF and B’D’KL. Dividing the rectangle along its diagonal BC gives two congruent 120-degree-angled triangles BD’C and BFC that have an area, by definition, of half the base length 𝑦=FC multiplied by its projected height NF=23x, or
Area of Triangle AB’B=21y(23x)=43xy
Concluding, all the area in Figure 9a of the smaller equilateral triangle FB’B associated with side 𝑥=B’B of area 43x2, the larger equilateral triangle EB’A associated with side 𝑦=B’A of area 43y2, plus the coupling 120-degree-angled triangle EB’F of area 43xy is accounted for in forming the longest side equilateral triangle ACB of area 43z2 in Figure 9b. The associated algebraic process of adding areas, leading to the extended version of the Pythagoras Theorem equation, follows below from the aforementioned description. Both Figure 9a and 9b represent independently in terms of area addition an equation, and both incorporate a central triangle that link both equations to provide the desire solution. In Figure 9a, the larger equilateral triangle EB’A of area 43y2 is composed of a smaller equilateral triangle HJI of area 43x2, two smaller skewed rectangles EHJD’ and IAA’J of combined area 22(y−x)(23x), plus the central triangle D’A’B’ of area 43(y−x)2, resulting in
that re-arranged gives
This forms one equation, and the second equation is retrieved from Figure 9b. This is now described. The equilateral triangle ACB in Figure 9b of area 43z2 is composed of three congruent 120-degree-angled triangles - AA’C, CD’B and BB’A - of area 43xy each, revolving around a central triangle A’D’B’ of area 43(y−x)2, giving
Subtracting Eq.(9) from Eq.(10) removes the linking term – the central triangle 43(y−x)2 – and Eq.(10) becomes
Expanding and simplifying the terms on the left, and placing 43y2 on the other side gives
Thus, completing the proof.
Conveniently removing the coefficient 43 results in the required relation
The initial challenge raised in Figure 2b regarding the extended Babylonian problem of the length of a cane against an inclined slope in Figure 2b (or an inclined wall) such that they form a 120-degree angle is answered by using Eq.(12) as z=y2+xy+x2, where the key difference with respect to the original orthogonal problem in Figure 2a having a solution z=y2+x2 is the coupling term 𝑥𝑦. While Eq.(12) is a more convenient way to write Eq.(11) as it speeds up calculations (and indeed the relative proportion between areas is expressed correctly), one must remember than in absolute terms the coefficient 43needs to be accounted for to express the actual area of the triangle 43x2 that is not 𝑥2, and thus the more complete (but less convenient) way to express the extended theorem is Eq.(11). Keeping this coefficient in mind, one can algebraically apply Eq.(12) as a matter of convenience.
As a summary, the classical Pythagoras Theorem interrelating squares via a right-angled triangle is shown in Figure 10a, while the extended Pythagoras Theorem interrelating triangles via a 120-degree-angled triangle is shown in Figure 10b. Building the extended version of the Pythagoras Theorem is almost as straight forward as the classical version, and only requires one to draw three equilateral triangles, one for each of the three sides of the 120-degree-angled triangle, and draw a line interconnecting the upper left edge of the 𝑦-bounded equilateral triangle to the lower left edge of the 𝑥-bounded equilateral triangle, to form the coupling area, as shown in Figure 10b.
As a conclusion, while the classical orthogonal Pythagoras Theorem interconnecting squares via a right-angled triangle (Figure 10a) is given by
The new extended Pythagoras Theorem interconnection of triangles via a 120 degree-angled triangle (Figure 10b) is given by
This result can be validated via the law of cosines by replacing the angle 𝛾=120° into Eq.(3), repeated below for convenience
which results in the substitution cos(120deg)=−21 giving
Thus arriving at the same result as Eq.(12)
Of particular interest is the coupling term 𝑥𝑦 that emerges as a key difference between the classical and the extended version of the Pythagoras Theorem. Its physical meaning will be explained in the following exercises.
Two classroom examples are presented with both theorems being applied, first relating dimensions of beams useful in bridge building for civil engineering, and second relating number of coils and their angular clocking used in the design of alternating current generators and motors for electrical engineering.
4.1 Civil Engineering
The following exercise, involving beams in a bridge (Figure 11), illustrates how both theorems are equally helpful. While this is a straight forward example, it is worthwhile recording it for the sake of those students that are beginners to the fundamental application of the Pythagoras Theorem, and its extensions. Imagine that one wants to know the distance between the arc and the platform at two different locations. At location A, the angle between the beams is orthogonal and the beams are 𝑥=0.7𝑚 and 𝑦=1𝑚 in length, and at location B the angle between the beams is 120 degrees and the beams are 𝑥=1.5𝑚 and 𝑦=2𝑚 in length. What are the respective distances?The answer is obtained by applying both the classical and extended Pythagoras Theorem. For the orthogonal case, the classical Eq.(1) gives
While for the case where both beams form 120 degrees, the extended Eq.(12) results in
4.2 Electrical Engineering
Systems inherently possessing orthogonality, like the pendulum, invoke the usage of the classical Pythagoras Theorem to determine its dynamical behavior. Similarly, systems operating based on 120-degree angles, like the electrical alternating current system, naturally invoke the usage of the extended Pythagoras Theorem using triangles, presented earlier in this article. As another associated example, three-phase electric power systems ― popularly employed in electrical engineering ― operate by setting three alternating currents that are out of phase by 120 degrees, while having the same frequency [10,11].
Alternating current generators and motors operate by placing coils around a rotating magnet. The electromagnetic field generated by the permanent magnet at the center is fixed, having a north and south pole where intensity is maximum. A gradual change in intensity is observed as one rotates the magnet from one pole to the other. This change in electromagnetic field is converted by a coil into alternating current. Naturally, having more coils produces more alternating current, but they have to be grouped around the magnet in a harmonious way. This is where the classical square-based Pythagoras Theorem and the new extended triangle-based version are useful. Figure 12 shows an alternating current motor/generator (the cut-out is for illustration purposes) with a zoom (on the right) to one of its sectors composed of two coils angular-spaced by 90º (assuming a quadruple coil system disposed as a square) with a rotating magnet at the center. How does one perceives the Pythagoras Theorem in a physical sense in an alternating generator or motor?
Imagine the peak electromagnetic field of the magnet is represented by the area of the square 𝑧2, and is naturally fixed and independent of angular position. As the magnet rotates, the intensity of the field going through coil A – here represented by the area of square 𝑥2 - is decreasing (in blue), while that in coil B – here represented by the area of the square 𝑦2 - is increasing (in purple). Therefore, the Pythagoras Theorem states that in an orthogonal system (i.e., when coils are clocked by 90 degrees), for every angle the magnet rotates, the drop in intensity in coil A (i.e., drop in area 𝑥2) is compensated by a raise in intensity in coil B (i.e., raise in area 𝑦2), or
Why is this important? Because the intensity of the magnetic field perceived by the system composed of the sum of the two coils is always the same, equal to the peak intensity of the magnet (i.e., the area 𝑧2), independently of the angular position of the magnet. That is, there is a continuous constant power generation (or loading in the case of a motor) as the magnet turns.
Exercise 1. Imagine the magnet has a hypothetical intensity of 1, and coil A is recording an intensity 𝑥2=0.72=0.49. How much intensity is being recorded by coil B located in a clockwise position at 90°, and how much is being wasted?
Solution. Assume 𝑧=12 and 𝑥2=0.72=0.49. Applying the classical Pythagoras Theorem gives
This means that coil A sees an intensity of 𝑥2=0.49 or 49%, while coil B sees 𝑦2=0.51 or 51%. Since the sum of the two gives 100% of the intensity of the magnet, nothing is being wasted. Figure 13 shows an alternating current motor/generator (the cut-out is for illustration purposes) with a zoom (on the right) to one of its sectors composed of two coils angular-spaced by 120º (assuming a triple-coil system disposed as a triangle) with a rotating magnet at the center.
This time, imagine the peak electromagnetic field of the magnet is represented by the equivalent area of the equilateral triangle 𝑧2, and is naturally fixed and independent of angular position. As before, when the magnet rotates, the intensity of the field going through coil A – here represented by the area of the square 𝑥2 - is decreasing (in blue), while that in coil B – here represented by the area of the square 𝑦2 - is increasing (in purple). However, because the coils are more angular-spaced apart, the rate at which the intensity changes between them will be different. The extended Pythagoras Theorem states that in a triple-phase system (i.e., when coils that are clocked by 120 degrees), for every angle the magnet rotates, the drop in intensity in coil A (i.e., drop in area 𝑥2) is compensated by a raise in intensity in coil B (i.e., raise in area 𝑦2) plus a part of this intensity (in yellow) that is not perceived by neither coil A and B (i.e., the coupling area 𝑥𝑦), giving
Again, why is this important? Because the total intensity of the magnetic field perceived by the system composed of these two coils drops (below the peak magnet intensity), as the magnet moves away from coil A until a minimum is reached halfway between the two coils, identifying a loss in the system. Beyond this, the system’s perceived intensity rises again until it reaches once more the peak intensity of the magnet (i.e., the area 𝑧2) at coil B. In this example, the extended Pythagoras Theorem using triangles not only tells us that the total electromagnetic intensity perceived by the system is not always constant, but also it quantifies how much of that intensity is wasted via the coupling term 𝑥𝑦.
Exercise 2. Imagine the magnet has a hypothetical intensity of 1, and coil A is recording an intensity 𝑥2=0.72=0.49. How much intensity is being recorded by coil B located in a clockwise position at 120°, and how much is being wasted?
Solution. Assume 𝑧=12 and 𝑥2=0.72=0.49. Applying the extended Pythagoras Theorem using triangles gives
Re-arranging gives the quadratic function
Finding the root gives
This means that coil A sees an intensity of 𝑥2=0.49 or 49%, while coil B sees only 𝑦2=0.198 or approximately 20%. For this particular angular position, the coupling part of the electromagnetic field generated by the magnet 𝑥𝑦=(0.7)(0.445)≈0.312 or approximately 31%, is not passing by either coil A or B, and is lost in a “blind spot” in an angular location between them. This 31% represents a waste, as the turning of the magnet is not fully electromagnetically inducting this sector of the system at this particular angular position. If one adds all the separate components, the result is the magnet peak intensity 1, or
To the operation of the present triple coil system disposed in a triangular manner, this unused electromagnetic induction capacity can be seen as a loss in efficiency, or simply a detriment that needs improving. The solution to this problem was introduced by Tesla that added an extra coil in between each of the original three, resulting in a hexagonal or dual-inverted triangle configuration . In turn, this hexagonal contemporary alternating current generator/motor is well known for delivering continuous power generation (or loading in the case of a motor) at constant rating for every magnet rotation, thus being used in industry all around the world throughout the past century until the present today.
Another example of an application in Electrical Engineering is a cellular base antenna station deployed in an array along three faces, arranged in a triangular manner ― with each face having typically 3 or 4 radiating elements (Figure 14a) , for which there are real examples (Figure 14b). The analysis done to the AC generator/motor is valid by replacing the rotating magnet with a transmitting mobile phone, and the coils by each face of the triangular antenna array. The directivity, and hence intensity of signal strength perceived by two of its three antenna array faces 1 and 2 disposed at 120 degrees to each other will vary as the phone rotates around the station (by the displacement of the individual, as it transits), recording the same reduction/increase in signal behavior as that explained before for the AC generator/motor application (where the magnet rotated at the center of two radially equidistant coils angular-spaced by 120 degrees).
The Pythagoras Theorem, while providing a fundamental cornerstone to science, is not singular. Like all great theories, the Pythagoras Theorem pertains to a larger family of possibilities. In this article, the classical approach of relating areas with squares is moved into the new arena of relating areas with triangles, which implies moving away from the right-angled triangle - as the connecting element - to a 120-degree-angled triangle. Both theorems are proven and discussed back-to-back. The classical Pythagoras Theorem is explained first, establishing a particular approach that then serves as a baseline to evolve into the new triangle-based theorem. A key outcome was the identification of the coupling term 𝑥𝑦 as the main difference between these two theorems. Being explained for the first time geometrically and mathematically, the origin of this coupling term reaches out to a time period before trigonometric functions (like cosine), where the law of cosines was not yet known. The usefulness of both the Pythagoras Theorem and it’s variant is demonstrated with dedicated classroom exercises on both civil engineering (bridge building) and electrical engineering (alternating current generators/motors and triangular cellular base antenna stations), showing that one theorem complements the other side-by-side, hinting that both are actually particular cases of a more general rule.
 Friberg, J. (1981). “Methods and traditions of Babylonian mathematics“. Historia Mathematica, 8, 227–318.
 Pickover, C. A. (2012) The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Co., Inc.
 Teia, L. (2021). “Extended Pythagoras Theorem Using Hexagons“. Canadian Center of Science and Education, Journal of Mathematics Research, 13(6), 46–51. Retrieved from https://doi.org/10.5539/jmr.v13n6p46 Accessed 14 January 2022.
I have decided to accept your paper for inclusion in The Journal of Open Engineering following your revisions in response to the reviewer comments.
It is clear that the author has a keen sense and love for geometrical proof, which is welcome. This article provokes thought about when an algebraic or analytical result can be proved by geometric means.
However, from an Engineering point of view, I regret to say that I do not see a lot of focus here. The examples from Civil and Electrical Engineering are at best of tangential interest, as the situations presented do not have any special affinity to the 120-degree angle, and so these examples provide no motivation for why this particular special case is important. There might be some examples in engineering with high degrees of symmetry that are more relevant, but even then, I cannot think of any situations in which knowing the direct result of the Law of Cosines for the 120-degree angle is anything more than a nice convenience in the presence of much more complex issues.
From an Engineering Education point of view, there is possible relevance - maybe it is useful to teach students to apply the 'short cut' of the Law of Cosine for the 120-degree case. But from my perspective having taught Statics for over 20 years, the students who struggle - and there are many - have deficiencies in algebra, geometry, and trigonometry. The proofs in this paper would escape the grasp of even most well-performing Statics students, and I think they would miss the central idea that ‘pure’ geometry can sometimes be used to derive analytical results. Moreover, emphasizing the single case for 120 degrees would distract from the more basic necessity, which is to help students learn how to confidently apply the Laws of Sine and Cosine in general cases.
From a historical perspective, there is some interesting Mathematics history to which the article alludes (see comments below), but I cannot think of anything analogous from the history of Engineering.
For the reasons stated below, I did not attempt to verify most of the equations. However, there is at least one important error, which occurs in equation (2): the (-) should be (+).
I found the proofs to be tortured and lengthy to the extent that I did not attempt to trace through most of the details. Instead, motivated by the author’s approach, I discovered a much-simplified proof based on adding a few line segments to the Figures 10(a) and 10(b); in Figure 10(b), the “z-triangle” can be inscribed in a larger equilateral triangle of sides x + 2y. I would be happy to discuss this with the author upon request.
There are numerous grammatical, typographical, and formatting errors, which I will not comment on explicitly, as I think there are more fundamental concerns with the article before reaching the point of editorial refinement. But one general comment is that all mathematical characters should be set in a standardized typeset, and all equations should be numbered.
The paper is complete in the sense that it presents an approach for deriving a result and completes a proof.
However, very problematic is at least one instance of plagiarism. The following segment is a direct phrase match from the reference :
“Thus, Euclid's Elements (300 B.C.) contains the seed of concepts that lead to the law of cosines. In the fifteenth century, the Persian astronomer and mathematician al-Kashi provided accurate trigonometric tables and expressed the theorem in a form suitable for modern usage.”
Although  is cited immediately following as a reference, there is no acknowledgment of the quotation. This MUST be corrected. I will also comment that this brief historical account, which jumps from Euclid to al-Kashi, is similar to the account in Wikipedia, whereas in , there is somewhat more history given. I have not studied this question in detail, but I would imagine that there is further history on the geometrical basis of the Law of Cosine that could be reported.
It is also both strange and inappropriate that the author did not cite another of his own, published works on an essentially similar topic, “Extended Pythagoras Theorem Using Hexagons” (https://ccsenet.org/journal/index.php/jmr/article/view/0/46329). I discovered this article because some near-phrase matches were detected from this article when I ran the manuscript through Ithenticate. Although the article on “ … Using Hexagons” uses different details to deduce the Law of Cosine for the different case of a 60-degree angle, vs. the 120-degree angle in the manuscript, the overall scheme is very similar, and requires citation.
OPENNESS AND REPRODUCIBILITY
This article appears to meet the openness and reproducibility standards. The references that I attempted to check were openly accessible (although I don’t think that cited works should be required to be open-access, despite the mission of this Journal).
I thank the reviewer for taking the time and effort to present his opinion on the content of this article, and for all the valuable comments.
I am afraid I do not agree with the reviewer on the first point, as there are several cornerstone examples of interest available. Three-phase electric power is popularly employed in electrical engineering, and operates by setting currents of combined common frequency systems out of phase by 120 degrees, and I quote (link directly below) “In electrical engineering, a three-phase system indicates a combined system of 3 alternating current circuits (for a system of production, distribution and consumption of electricity) that have the same frequency.” A sentence was added at the beginning of section 4.2 to acknowledge this, along with an appropriate reference.
Another example of an important application in electrical engineering is telecommunications, in particular cellular base station antennas deployed in an array along three faces arranged in a triangular manner ― with each face having typically 3 or 4 radiating elements (see link below on page 336; and the second link for a photo of a real triangular antenna station). The analysis done to the AC generator/motor is valid by replacing the rotating magnet with a transmitting mobile phone, and the coils by each face of the triangular antenna array. The directivity, and hence intensity of signal strength perceived by two of its three antenna faces disposed at 120 degrees to each other will vary as the phone rotates around the antenna (by the displacement of the individual, as it transits), recording the same reduction/increase in signal behavior as that explained before for the AC generator/motor application (where the magnet rotated within two coils angular-spaced by 120 degrees). This has been added to section 4.2.
Favoring complex issues over simplicity is ― in this author’s opinion ― the first step to overlooking the great discoveries awaiting when adopting a simple approach to investigate any given challenge. Here the book “Small is beautiful” conveys the right philosophical message herein intended.
Moreover, the simple fact that the present article determines that the variant equation interconnects triangles, is in itself something that was not known (let alone proven) by simply applying the law of cosines, in its present form and understanding. The present understanding of the law of cosines is incomplete, as the geometrical linkage between the areas in the equation is not known or proven for any angle gamma, except for 90 degrees in the Pythagoras theorem connecting squares (with its own various geometrical and mathematical proofs), and now for 60 and 120 degrees in the new variants of this theorem connecting triangles and hexagons. I leave open the challenge of extending these proofs for gamma equal to 108 degrees, which interconnects the area of pentagons.
Throughout science, it is known that studying fundamental systems is the precursor to perceiving applications to more complex end situations. One cannot do an application without first formulating the theory (the basis for any PhD thesis). The present case is no exception, and the first step is completely fulfilled by presenting the novel theory using triangles. Then, it provides a potential application to improve the understanding of fundamental systems such as AC motor/generator and triangular antenna arrays, where no comments were provided invalidating the present logic and conclusions. Other more complex end applications are out of the scope of this article.
Another simple example in history where theory precedes application is the Taylor series that was invented in the 18th century by Brook Taylor, and found centuries later many of its most critical applications in physics ― such as Numerical Analysis & Methods (for example the Navier-Stokes equations used in Computational Fluid Dynamics), Quantum & Nano Physics and Calculators.
Not to mention Einstein’s Theory of Relativity that was opposed by many preeminent scholars by many years as being wrong (let alone it being useful with an application), and now it is the cornerstone of modern physics. Hence, in the authors perspective, not finding an application simply is a synonym of not trying hard enough, while (in this case) at the same time, of also not knowing enough to do so.
Another application of the present theory that is foreseen by this author in the future ― but it is too extensive and merits an article of its own ― is in electronics. The RLC circuit operates based on the principle of the Pythagoras theorem (via the same dynamics that govern a mass-spring-damper system), hence it is possible to conceive a variant circuit that operates based on the new theorem, something that this author has named coupled-RLC circuit. Again, this is an application that the reviewer cannot know about in the open literature of electrical engineering, simply because this author has not yet published it.
This article does not intend to teach students of existing trigonometric knowledge (that is not its purpose), as this is well covered by existing literature. It is the intention of this article to push the boundaries of the known trigonometry into the unknown, and that may be discomforting and disorientating to anyone uninitiated to such a new topic ― whether it be a student or a teacher. The fact that teachers spend decades teaching the same thing, sure compels and welcomes a distraction with a novel fresh idea into the possible future of trigonometry. Naturally, nothing new comes from doing the same thing repeatedly. To this respect, the present work is highly necessary to both mathematical research and teaching, and its natural subsequent application to engineering.
A common problem found in mathematical proofs is the lack in clarity of the steps taken (a struggle that mathematical genius Ramanujan highlighted with his own life and work experience). Hence at the expense of extra attention to detail, this proof is published with high precision. This obviously requires more length in explanation, and consequentially more attention from the reader, which is a good thing as it compels her/him to expand its own capabilities to interpret the inherent complexities of mathematics. Further simplified versions of this proof are possible for popularized explanations ― like for teaching ― in the future to come, however the initial proof needs to be as precise as possible for completeness in capturing the totality of the proof.
If the reviewer has found a variant of this proof, the best venue would be for him to submit it to a peer-reviewed journal publication, of which I reciprocally would be happy to be a reviewer, and assess the truth and usefulness of its content.
Referring back to the need for precision and proofs, if these mentioned grammatical, typographical and formatting errors are not highlighted by the reviewer (due to a matter of convenience), then there is no proof that they exist. This is strange, as the article was initially written in MS Word which has a spell checker, passed to TJOE which also has its own spell checker with a semi-automatic HTML text formatting, and the other reviewer (who demonstrated to be quite thorough) only highlighted a few typos that are already corrected. Given the low input, the most that can be done is to pass the article again through a spell checker (no errors were mentioned from the actual mathematical and geometrical content), and see if something has been overlooked. Upon conclusion of this task, nothing new was detected.
The sentence in question ― the result of a simple overlook ― was paraphrased and source cited to meet the referencing standards of MIT online publication on academic integrity (link below) .
As for the Wikipedia comment, by doing the appropriate investigation, one can find that the source of the quoted statement ― which is presented both in this article and in the Wikipedia page ― is referenced to the book from Clifford A. Pickover, which is properly identified in both this article (as reference ) and in Wikipedia. The amount of history provided on the law of cosines is deemed sufficient for the purpose of this article. Hence, introducing a further statement that more history exists on the law of cosines, without making a proper investigation and/or referencing, can only be regarded as inconclusive speculation.
The reason why my most recent work “Extended Pythagoras Theorem Using Hexagons” was not yet referred is simple. Noticing back to its release date (November 21, 2021), it is seen that it is later than that of the release of the present draft (November 15, 2021), hence the first not appearing in the second. Being of equal importance and relevance to the present work, the hexagonal proof is now remarked (as originally intended by this author) as an added sentence at the end of section 1.2, and appropriately referenced in the final version of this publication as reference .
Thank you for acknowledging that the present article meets the standards of this journal.
I would like to thank once again the reviewer for his time and effort in contributing to the assessment of this article, and I wish him a prosperous future.
Luis Teia, PhD
I have completed answering your recommendations to the best of my abilities (the answers were posted in the original article, while the rectifications were issued as a new draft - see links below). Please have a look to see if there is any point you wish me to further elaborate. I have tried to be methodical.
It was a pleasure working with you. Kind regards, Luis Teia
I don’t think it’s incorrect, just not what you’ve gone with for your proof - which is focussed on equilateral triangles rather than squares (or any other shape).
To avoid confusion, the sentence was removed (in the new draft version). Cheers.
Proof has been completed when you get to (11). Agree that (12) is more convenient, but not the theorem you have set up.
Agreed. I have placed (in the new draft of the article) both the phrase “Conveniently…” and Eq.(12) below the horizontal line, resulting in the completion of the proof just below Eq.(11). Thanks
Should this be prior to the previous paragraph? Seems out of order to be using this in a formula before establishing the area of these 120-degree triangles.
Yes, I see your point and agree. The explanation of the area of the 120-degree triangles was moved up (in the new draft version) to just before “Concluding…”, just after the sentence “The 120-degree-angled triangle in Figure 9a EB’F becomes AB’B in Figure 9b.“ This way it is in the flow of the explanation, and before using it in any of the formulas. Cheers!
Term added (in the new draft)
Is Figure 8 intentionally drawn so that triangles KML and FLB appear approximately equal?
Not really, this was a coincidence. To avoid confusion, Figure 8 was slightly altered (i.e., triangle ABC was slightly rotated, with the remainder triangles and clockwork following this rotation) [in the new draft of this article] so that the difference between KML and FLB is more noticeable (i.e., generally they are different). This rotation is minor, and does not affect the validity of any of the explanations and arguments given.
I understand hypotenuse to only have meaning in the context of a right-angled triangle. Perhaps “longest side” would babe more appropriate?
The recommended correction was made (in the new draft) for all instances related to the triangle version of the theorem. Cheers.
Given the non-standard axes in use here, would it be worth stating that you’re setting up three axes, x, y and w at 60 degrees to one another? This will make establishing that triangles are equilateral more watertight.
Indeed, an early warning would facilitate the reader’s transition between theorems. I have added a paragraph (in the new draft article) just after the title of Chapter 3, for this specific purpose. Thanks.
How do we know this is equilateral? (I’m convinced that they are, but only intuitively.)
I see the concern. To resolve this, an explanation was added to the text (just before this sentence, and in the new draft version of this article), describing step-by-step the arguments that, emerging directly from the extension of segments parallel to each axis from the corners of the outer equilateral triangle DAB, these define amongst themselves the triangle KML as being equilateral. Cheers
Term added (in the new draft)
The tense doesn’t match the surrounding writing - “changes” instead
Tense changed as suggested (in the new draft)
It seems strange in Figure 5 to have the orthogonal axes positioned diagonally rather than vertical/horizontal, and it isn’t immediately obvious why it has been done this way (it makes sense after seeing Fig 7, but I’m still not convinced that it is the best way to represent this)
It’s a good point ! To make it easier for all to understand, the orthogonal axes in Figure 5 and 6 were altered back (in the new draft version of this article) to the traditional vertical and horizontal layout. An explanation is later introduced determining how these axes alter (in number and place), when we move to the triangular version of the theorem.
This doesn’t sound quite right - may be a translation issue?
This point was rewritten and further elaborated (in the new version of the article), for added clarity.
In Figure 6, below, it might be worth adjusting the rotation so that it is clear that point N’ is not on the circle.
Figure 6 has been changed to make it clear that point N’ is (in general) not on the circle. Cheers
Previously sides x and y were defined as the distances from A and B to O, which happened to be coincident with the orthogonal axes. It may be better to define these lines initially as being parallel to the orthogonal axes, and meeting at a point M, and then because in Figure 5 these lines are coincident with the orthogonal axes, then M=O.
Yes, I see your point and agree. I have altered (in the new draft of this article) the nomenclature such that the sides x and y for both theorems are related to distances from A and B to M — both parallel to the axes — for which the particular case x equal y coincides point M with the origin of the axes given by Point O — i.e., M=O — and for the general case x different from y, point M is not the same as point O. Thanks