Thales' theorem

Thales' Theorem states that if there are three points on a circle, $A, B, C$ with $AC$ being a diameter, $\angle ABC=90^{\circ}$ .

$[asy] dot((5,0)); dot((-5,0)); draw(circle((0,0),5)); dot((3,4)); draw((5,0)--(3,4)); draw((-5,0)--(3,4)); draw((-5,0)--(5,0)); label("A",(-5,0),W); label("B",(3,4),NE); label("C",(5,0),E); [/asy]$

This is proven by considering that the intercepted arc is a semicircle and has measure $180^{\circ}$ . Thus, the intercepted angle has degree measure $\frac{180}{2} = 90$ .

This theorem has many uses in geometry because it helps introduce right angles into problems; however, the name of the theorem is not well-known. Thus, you may cite the "universal fact" that $\angle ABC = 90^{\circ}$ in proofs without specifically referring to Thales.

Problems

1. Prove that the converse of the theorem holds: if $\angle ABC = 90^{\circ}$ , $AC$ is a diameter.

2. Prove that if rectangle $ABCD$ is inscribed in a circle, then $AC$ and $BD$ are diameters. (Thus, $AC = BD$ .)

3. $AC$ is a diameter to circle O with radius 5. If B is on O and $AB = 6$ , then find $BC$ .

4. Prove that in a right triangle with AD the median to the hypotenuse, $AD = BD = CD$ .

5. $AC$ is a diameter to circle O, B is on O, and D is on the extension of segment $BC$ such that $AD$ is tangent to O. If the radius of O is 5 and $AD = 24$ , find $AB$ .

6. In a triangle $ABC$ , $CD$ is the median to the side $AB$ ( $D$ is the midpoint). If $CD=BD$ , then prove that $\angle C=90^\circ$ 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.

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Thales' theorem

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