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PAMO 2022 Problem 1 - Line Tangent to Circle Through Orthocenter
Source: 2022 Pan-African Mathematics Olympiad Problem 1
6/25/2022
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
B
C
≠
9
0
∘
\angle ABC \neq 90^\circ
∠
A
BC
=
9
0
∘
, and
A
B
AB
A
B
its shortest side. Let
H
H
H
be the orthocenter of
A
B
C
ABC
A
BC
. Let
Γ
\Gamma
Γ
be the circle with center
B
B
B
and radius
B
A
BA
B
A
. Let
D
D
D
be the second point where the line
C
A
CA
C
A
meets
Γ
\Gamma
Γ
. Let
E
E
E
be the second point where
Γ
\Gamma
Γ
meets the circumcircle of the triangle
B
C
D
BCD
BC
D
. Let
F
F
F
be the intersection point of the lines
D
E
DE
D
E
and
B
H
BH
B
H
.Prove that the line
B
D
BD
B
D
is tangent to the circumcircle of the triangle
D
F
H
DFH
D
F
H
.
geometry
circumcircle