1. Prove the Pythagorean Theorem by a method not used in class..

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