1960 IMO Problems/Problem 3
In a given right triangle , the hypotenuse , of length , is divided into equal parts ( an odd integer). Let be the acute angle subtending, from , that segment which contains the midpoint of the hypotenuse. Let be the length of the altitude to the hypotenuse of the triangle. Prove that:
Using coordinates, let , , and . Also, let be the segment that contains the midpoint of the hypotenuse with closer to .
Then, , and .
So, , and .
Since , and as desired.
Let be points on side such that segment contains midpoint , with closer to and (without loss of generality) . Then if is an altitude, then is between and . Combined with the obvious fact that is the midpoint of (for is odd), we have
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