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Incenter, Excenter and intersections

Source: Pan-American Girls’ Mathematical Olympiad 2021, P6

October 6, 2021

geometryincenterexcenterPAGMO

Problem Statement

Let

ABC

be a triangle with incenter

I

, and

A

-excenter

\Gamma

. Let

A_1,B_1,C_1

be the points of tangency of

\Gamma

with

BC,AC

and

AB

, respectively. Suppose

IA_1, IB_1

and

IC_1

intersect

\Gamma

for the second time at points

A_2,B_2,C_2

, respectively.

M

is the midpoint of segment

AA_1

. If the intersection of

A_1B_1

and

A_2B_2

is

X

, and the intersection of

A_1C_1

and

A_2C_2

is

Y

, prove that

MX=MY

.

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