Problem 52: Iran MO 1996, 3rd round, 3rd exam, Problem 2
by henderson, Oct 1, 2016, 11:45 AM
Consider a semicircle of center 
and diameter 
A line intersects 
at 
and the semicircle at 
and 
such that 
and 
The circumcircles of 
and 
intersect again at 
Prove that 
Denote by 
and 
the circumcircles of 
and the quadrilateral 
respectively. Then, 
and 
Thus, by the 
and 
are concurrent. Let 
be their intersection point.
It's easy to see that 
and 
is the orthocenter of 
Hence 
Combining this fact with 
is a cylic quadrilateral 
is a cyclic quadrilateral, too 
is a cyclic pentagon. Therefore, 
and 
as desired.
(Iran MO 1996, 3rd round)
This post has been edited 8 times. Last edited by henderson, Oct 21, 2016, 1:51 PM
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