AoPS - Problem

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Problem

Source: JBMO Shortlist 2013, G4

Tags: incenter, circumcircle, excircle, geometry, perpendicular

Steff9

11.06.2017 13:32

Let $I$ be the incenter and $AB$ the shortest side of the triangle $ABC$ . The circle centered at $I$ passing through $C$ intersects the ray $AB$ in $P$ and the ray $BA$ in $Q$ . Let $D$ be the point of tangency of the $A$ -excircle of the triangle $ABC$ with the side $BC$ . Let $E$ be the reflection of $C$ with respect to the point $D$ . Prove that $PE\perp CQ$ .

andrei.pantea

11.06.2017 14:07

Consider

$IX\perp BC$ and

$IY\perp AB$ .

Now it should be easy to see that

$BE=BP$ , and we're done by angle chasing.

jayme

11.06.2017 15:30

Dear Mathlinkers, see also http://www.artofproblemsolving.com/community/c6t48f6h1097017_nice_geometry_problem Sincerely Jean-Louis

sttsmet

29.04.2021 12:05

Can someone gives us a good shape, because I still havent understood what is the A-excircle!