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Incenters, Excenters, and Feet, OH MY!

Source: 2016 USAMO 3

April 19, 2016

geometry2016 USAMOUSAMOUSA(J)MO

Problem Statement

Let

\triangle ABC

be an acute triangle, and let

I_B, I_C,

and

O

denote its

B

-excenter,

C

-excenter, and circumcenter, respectively. Points

E

and

Y

are selected on

\overline{AC}

such that

\angle ABY=\angle CBY

and

\overline{BE}\perp\overline{AC}

. Similarly, points

F

and

Z

are selected on

\overline{AB}

such that

\angle ACZ=\angle BCZ

and

\overline{CF}\perp\overline{AB}

.
Lines

\overleftrightarrow{I_BF}

and

\overleftrightarrow{I_CE}

meet at

P

. Prove that

\overline{PO}

and

\overline{YZ}

are perpendicular.
Proposed by Evan Chen and Telv Cohl

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