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Germany Team Selection Test
2023 Germany Team Selection Test
2
2
Part of
2023 Germany Team Selection Test
Problems
(1)
Orthocenter, Tangent points and circumcircles
Source: German TST 2023 AIMO 5, Problem 2
11/2/2023
Let
A
B
C
ABC
A
BC
be an acute angled triangle with orthocenter
H
H
H
and
A
B
<
A
C
AB<AC
A
B
<
A
C
. The point
T
T
T
lies on line
B
C
BC
BC
so that
A
T
AT
A
T
is a tangent to the circumcircle of
A
B
C
ABC
A
BC
. Let lines
A
H
AH
A
H
and
B
C
BC
BC
meet at point
D
D
D
and let
M
M
M
be the midpoint of
H
C
HC
H
C
. Let the circumcircle of
A
H
T
AHT
A
H
T
meets
C
H
CH
C
H
in
P
≠
H
P \not=H
P
=
H
and the circumcircle of
P
D
M
PDM
P
D
M
meet
B
C
BC
BC
in
Q
≠
D
Q \not=D
Q
=
D
. Prove that
Q
T
=
Q
A
QT=QA
QT
=
Q
A
.
geometry
circumcircle