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Reflections of AB, AC with respect to BC and angle bisector of A

Source: BMO 2024 Problem 1

April 29, 2024

geometrygeometric transformationreflectionangle bisector

Problem Statement

Let

ABC

be an acute-angled triangle with

AC > AB

and let

D

be the foot of the

A

-angle bisector on

BC

. The reflections of lines

D

and

AC

in line

BC

meet

AC

and

AB

at points

E

and

F

respectively. A line through

D

meets

AC

and

AB

at

G

and

H

respectively such that

G

lies strictly between

A

and

C

while

H

lies strictly between

B

and

F

. Prove that the circumcircles of

\triangle EDG

and

\triangle FDH

are tangent to each other.

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