1994 IMO Problems/Problem 2
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Let 
be an isosceles triangle with 
. 
is the midpoint of 
and 
is the point on the line 
such that 
is perpendicular to 
. 
is an arbitrary point on different from 
and 
. 
lies on the line and 
lies on the line 
such that 
are distinct and collinear. Prove that 
is perpendicular to 
if and only if 
.
Solution
Let 
and 
be on and respectively such that 
. Then, by the first part of the problem, 
. Hence, is the midpoint of and 
, which means that 
is a parallelogram. Unless 
and 
, this is a contradiction since 
and 
meet at 
. Therefore, and , so 
, as desired.
