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simple geometry

Source: 2024 Girls in Mathematics Tournament, Level A, Problem 3

October 26, 2024

geometrysharky devil point

Problem Statement

In a triangle scalene

ABC

, let

I

be its incenter and

D

the intersection of

AI

and

BC

. Let

M

and

N

points where the incircle touches

AB

and

AC

, respectively. Let

J

be the second intersection of the circumcircle

TI

with the circumcircle

(ABC)

. Let

T

the intersection of

AF

and

BC

. Let

J

be the intersection of

TI

with the line parallel of

FI

that passes through

D

. Prove that the line

AJ

is perpendicular to

BC

.

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