1998 APMO Problems/Problem 4
From AoPSWiki
Problem
(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 
.
Solution
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.
