Extended Pythagoras Theorem using Triangles, and its Applications to Engineering

Engineering is governed by mathematical laws, of which a pearl is the Pythagoras Theorem - 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 . Here, it was found that just as the Pythagoras Theorem establishes an area relation between squares along the sides of a right-angled triangle, equation 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.


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 to a vertical column of height via intermediate diagonal beams of length ( Figure 1a). 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.

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) [1]. 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 2 start by assuming the simple case where the length of the cane along the ground to the wall is 3 meters, i.e., =3, and the length of the cane along the wall from the ground is 4 meters, i.e., =4. The Pythagoras Theorem gives us the direct answer 3 2 +4 2 =5 2 , resulting in a cane length of 5 meters, i.e., =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 [2], the relation between the area of the three squares surrounding an obtuse triangle ( Figure 3) is expressed as 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 [3]. Replacing where 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 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.
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 and to find the solution to the unknown diagonal side .
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 [4].

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 [5].
Theorem 1 (Classical Pythagoras Theorem using Squares). If x, y and z are real numbers, then the lengths of the sides and 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 [6]. 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 ≠ , the split of the combined area of squares 2 and 2 is shown to fit completely in the square of 2 . This is now further explained. From Figure 4a, the square EAOF associated with the side can be decomposed into the central square KLOB', the square EPKQ side , plus the two side rectangles PALK and QKB'F of area ( −��) each.  The proof is complete.

Particular case
The particular case of = 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 [7]. 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 = , 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 -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). Geogebra [9].
It is seen in Figure 5 that the four right-angled triangles -FEA, FMA, GMB and GHBforming the two smaller squares FEAM and GMBH with a combined area of 2 + 2 , are

General case
Moving now to the general case ≠ , using as foundation the former particular case = , imagine the larger square ACDB in Figure 5 of area 2 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  Algebraically, the squares JMBH' and IEAM give a combined area of 2 + 2 that has been shown in Proof 1 to be equal to the area 2 of square ACDB, thus concluding in the relation Figure 6. Classical (square-based) Pythagoras Theorem for the general case x≠y Figure 6 is the foundation from which the general case ≠ of the triangle version of the Pythagoras Theorem will be derived.

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 yaxis) 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.

Particular case
Assume the length of the smaller sides and of the 120-degree-angled triangle are the same. Also, for this particular case = , assume the prior square FABG in Figure 5 is now replaced by an equilateral triangle DAB in Figure   Where the coefficient can be conveniently cancelled on either side, resulting in

General case
Imagine the outer equilateral triangle ACB of area 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 (inside the circle) displaces along its edges, making the reference 120-degree-angled triangle change size, shape and position. For the general case ≠ , the now scalene 120-degree-angled triangle AMB in Figure 8,    Conveniently removing the coefficient 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 , where the key difference with respect to the original orthogonal problem in Figure 2a having a solution is the coupling term . 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 needs to be accounted for to express the actual area of the triangle that is not 2 , 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 rightangled triangle is shown in Figure 10a, while the extended Pythagoras Theorem interrelating triangles via a 120-degree-angled triangle is shown in Figure 10b.  Of particular interest is the coupling term 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.

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.

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 3.041 …

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   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. 0.714 … Figure 13. Extended Pythagoras Theorem application to a sector of an electric motor/generator with 120º-spaced coils 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 2 ) is compensated by a raise in intensity in coil B (i.e., raise in area 2 ) plus a part of this intensity (in yellow) that is not perceived by neither coil A and B (i.e., the coupling area ), 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 2 ) 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 .

Exercise 2.
Imagine the magnet has a hypothetical intensity of 1, and coil A is recording an intensity 2 =0.7 2 =0.49. How much intensity is being recorded by coil B located in a clockwise position at 120°, and how much is being wasted?  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 [12]. 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) [13], 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).

Conclusion
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 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