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Configurational Incenter-Excenter Geometry

Source: Indonesian Stage 1 TST for IMO 2022, Test 4 (Geometry)

December 25, 2021

Fact 5geometry

Problem Statement

In a nonisosceles triangle

ABC

, point

I

is its incentre and

\Gamma

is its circumcircle. Points

E

and

D

lie on

\Gamma

and the circumcircle of triangle

BIC

respectively such that

AE

and

ID

are both perpendicular to

BC

. Let

M

be the midpoint of

BC

,

N

be the midpoint of arc

BC

on

\Gamma

containing

A

,

F

is the point of tangency of the

A-

excircle on

BC

, and

G

is the intersection of line

DE

with

\Gamma

. Prove that lines

GM

and

NF

intersect at a point located on

\Gamma

.
(Possibly proposed by Farras Faddila)

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