Thales' Theorem states that if there are three points on a circle, with being a diameter, .
This is easily proven by considering that the intercepted arc is a semicircle, or 180°. Thus, the intercepted angle is 180°/2 = 90°.
This theorem has many uses in geometry because it helps introduce right angles into a problems; however, the name of the theorem is not well-known. Thus, you may cite the "universal fact" that <ABC = 90° in a proof without specifically referring to Thales.
1. Prove that the converse of the theorem holds: if , is a diameter.
2. Prove that if rectangle is inscribed in a circle, then and are diameters. (Thus, .)
3. is a diameter to circle O with radius 5. If B is on O and , then find .
4. Prove that in a right triangle with AD the median to the hypotenuse, .
5. is a diameter to circle O, B is on O, and D is on the extension of segment such that is tangent to O. If the radius of O is 5 and , find .
6. In a triangle , is the median to the side ( is the midpoint). If , then prove that without using Thales' theorem. If you have a general understanding of how the theorem works and its proof you can manipulate it into the solution.
Please add more problems!