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Innovative Approaches to Proving the Pythagorean Theorem and Related Geometric Theorems

This article explores various methods to prove the Pythagorean Theorem that are not typically taught in classrooms. By constructing semicircles, golden rectangles, and equilateral triangles on the sides of a right triangle, we examine the areas of these geometric shapes and their relationships to the theorem. Additionally, we delve into Ceva’s and Menelaus’s Theorems to establish the concurrency of medians and angle bisectors in triangles. These methods not only demonstrate the theorem's validity but also promote deeper understanding and exploration of geometry beyond traditional methods.

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Innovative Approaches to Proving the Pythagorean Theorem and Related Geometric Theorems

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  1. § 12.1 1. Prove the Pythagorean Theorem by a method not used in class.. There are over 260 of them. You should not have had too much trouble finding another one.

  2. 2. On the three sides of a right triangle construct semicircles with centers at the midpoints of the sides. Calculate the area of each of the three semicircles. Do you see a relationship? c a b Do you think it works for other geometric figures?

  3. 3. On the three sides of a right triangle construct golden rectangles. Calculate the area of each of the three rectangles. Do you see a relationship? 3 0.61803 b Area of rectangle 1 = 0.61803 a 2 c 2 b 0.61803 c Area of rectangle 2 = 0.61803 b 2 a 1 0.61803 a Area of rectangle 3 = 0.61803 c 2

  4. 4. On the three sides of a right triangle construct equilateral triangles. Calculate the area of each of the three triangles. Do you see a relationship? 3 Area of triangle 1 = 0.4330 a 2 c 2 b Area of rectangle 2 = 0. 4330 b 2 a 1 Area of rectangle 3 = 0. 4330 c 2

  5. ad – af = bc - be a (d – f) = b (c – e) 5. Theorem

  6. 6. Use Ceva’s Theorem to prove that the medians of a triangle concur. A N M C B L AN = NB, BL = LC and CM = MA by definition of median. And by Ceva since the ratio is 1 the medians concur.

  7. 7. Use Menelaus’ Theorem in triangle ABE to prove that medians BE and CF meet at G, the two-thirds point on BE from B to E. A F E G C AF = FB, AE = EC by definition of median. B Consider ABE with points F, G, and C collinear. By Menelaus’ Theorem

  8. 8. Using the property c/b = a 1/a 2for angle bisectors (in the figure, Ad is the bisector of CAB and BD = a 1, DC = a 2), use Ceva’s Theorem to prove that the angle bisectors of a triangle are concurrent. A c 1 b 2 F E c 2 I b 1 And by Ceva since the ratio is 1 the angle bisectors concur. a 1 a 2 B D C

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