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Looks like some madness geo again

Source: European Mathematical Cup 2022, Junior Division, Problem 3

December 19, 2022

geometryincircleParallel Linesangle bisector

Problem Statement

Let

ABC

be an acute-angled triangle with

AC > BC

, with incircle

\tau

centered at

I

which touches

BC

and

AC

at points

\angle ACB

and

E

, respectively. The point

M

on

\tau

is such that

BM \parallel DE

and

M

and

B

lie on the same halfplane with respect to the angle bisector of

\angle ACB

. Let

F

and

H

be the intersections of

\tau

with

BM

and

CM

different from

M

, respectively. Let

J

be a point on the line

AC

such that

JM \parallel EH

. Let

K

be the intersection of

JF

and

\tau

different from

F

. Prove that

ME \parallel KH

.

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