Problem 48: The incenter lies on circumcircle
by henderson, Sep 14, 2016, 12:46 PM
Let 
be an acute triangle and let 
be a line in the plane of triangle 
We've drawn the reflection of the line over the sides 
and 
and they intersect in the points 
and 
Prove that the incenter of the triangle 
lies on the circumcircle of the triangle 
Let us assume that 
at 
and the extension of 
at 
become internal bisectors of 
and 
is the incenter of 
is the internal bisector of 
and 
become the external bisector and the internal bisector of 
and 
respectively 
is the 
-
is the internal bisector of 
Therefore, 
is the incenter of 
From the same fact that 
it follows that 
and 
Then 
and the result follows.
(Iran MO 1995, Round 2, Problem 2)
This post has been edited 7 times. Last edited by henderson, Sep 28, 2016, 12:19 PM
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