1998 APMO Problems/Problem 4
(Răzvan Gelca) Let be a triangle and the foot of the altitude from . Let and be on a line through such that is perpendicular to , is perpendicular to , and and are different from . Let and be the midpoints of the line segments and , respectively. Prove that is perpendicular to .
We use directed angles mod .
Since and are both right angles, points are concyclic. It follows that Similarly, the quadrilateral is cyclic, so Thus and are similar triangles, so and are similar figures. It follows that so points are concyclic. Since is a right angle, it then follows that is a right angle, as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
- 1989 APMO Problems
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