Problem 15: The converse of the butterfly theorem works here
by henderson, Jun 1, 2016, 11:58 AM
The circle 
passing through the vertices 
and 
of a triangle 
with 
intersects the sides 
and 
at the points 
and 
respectively. Let 
be the midpoint of 
The line perpendicular to 
at 
intersects 
and 
at 
and 
respectively. If 
prove that 
Applying the converse of to the quadrilateral 
be a circle of center 
and 
be four arbitrary points on this circle. Denote by 
and by 
the line passing through 
If 
and 
then the point 
we conclude that 
where 
is the circumcenter of the quadrilateral 
Since 
it follows that 
If 
then 
holds 
because of 
and 
so 
which is impossible. Therefore, and coincide. This leads to 
This post has been edited 7 times. Last edited by henderson, Mar 27, 2017, 2:45 PM
May 2017
March 2017