1965 AHSME Problems/Problem 30
Let of right triangle be the diameter of a circle intersecting hypotenuse in . At a tangent is drawn cutting leg in . This information is not sufficient to prove that
We will prove every result except for .
By Thales' Theorem, and so . and are both tangents to the same circle, and hence equal. Let . Then , and so . We also have , which implies . This means that , so indeed bisects . We also know that , hence . And as .
Since all of the results except for are true, our answer is .
It's easy to verify that always equals . Since changes depending on the sidelengths of the triangle, we cannot be certain that . Hence our answer is .