Whoever guesses the pun gets a prize
by shiningsunnyday, Sep 25, 2016, 6:08 PM
Vietnam TST 2001 wrote:
In the plane, two circles intersect at
and
and a common tangent intersects the circles at
and
Let the tangents at
and
to the circumcircle of triangle
intersect at
and let
be the reflection of
across the line
Prove that the points
are collinear.












![[asy]
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[/asy]](http://latex.artofproblemsolving.com/a/8/0/a80741dc5505597182f8338bac8bd65195278f47.png)
This is one of those cases where prior knowledge of a configuration/lemma instantly kills the problem.
Lemma:

Since




By a common characterization of the symmedian, we know that












Tidbit
This was kinda easy, mainly because I originally planned on posting USAMO 2008 P2 (which I've yet to solve DARN). Anyways, since these cyclic quads angle chasing stuff blah blah are getting kinda repetitive, the next few geo posts will be projective, which was my favorite topic of Geo 3 - cause it feels so freaking good using it to nuke the entire Euclidean plane LOL.
Also this problem's diagram looks pretty funny (can you see what it is?), part of why I posted this prob.
Also this problem's diagram looks pretty funny (can you see what it is?), part of why I posted this prob.
This post has been edited 1 time. Last edited by shiningsunnyday, Sep 25, 2016, 6:38 PM