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Orthocenter, Tangent points and circumcircles

Source: German TST 2023 AIMO 5, Problem 2

November 2, 2023

geometrycircumcircle

Problem Statement

Let

ABC

be an acute angled triangle with orthocenter

H

and

AB<AC

. The point

T

lies on line

BC

so that

AT

is a tangent to the circumcircle of

ABC

. Let lines

AH

and

BC

meet at point

D

and let

M

be the midpoint of

HC

. Let the circumcircle of

AHT

meets

CH

in

P \not=H

and the circumcircle of

PDM

meet

BC

in

Q \not=D

.
Prove that

QT=QA

.

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