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Challenging geometry with incircles

Source: Bulgaria EGMO TST 2017 Day 1 Problem 2

February 3, 2023

geometryincenterincircleexcirclecircumcirclegeometric transformationreflection

Problem Statement

Let

ABC

be a triangle with incenter

I

. The line

AI

intersects

BC

and the circumcircle of

BC

at the points

T

and

S

, respectively. Let

K

and

M

be the incenters of

T

and

SCT

, respectively,

M

be the midpoint of

BC

and

P

be the reflection of

I

with respect to

KL

. a) Prove that

M

,

T

,

K

and

L

are concyclic. b) Determine the measure of

\angle BPC

.

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