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Equal distance to triangle vertices

Source: Spain MO 2023 P6

March 12, 2023

geometrySpain

Problem Statement

In an acute scalene triangle

ABC

with incenter

I

, the line

AI

intersects the circumcircle again at

D

, and let

J

be a point such that

D

is the midpoint of

IJ

. Consider points

E

and

F

on line

BC

such that

IE

and

JF

are perpendicular to

AI

. Consider points

G

on

AE

and

H

on

AF

such that

IG

and

JH

are perpendicular to

AE

and

AF

, respectively. Prove that

BG=CH

.

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