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National and Regional Contests
Belarus Contests
Belarus Team Selection Test
2024 Belarus Team Selection Test
2.3
2.3
Part of
2024 Belarus Team Selection Test
Problems
(1)
XQ passes through orthocenter
Source: Belarus TST 2024
7/17/2024
A right triangle
A
B
C
ABC
A
BC
(
∠
A
=
90
\angle A=90
∠
A
=
90
) is inscribed in a circle
ω
\omega
ω
. Tangent to
ω
\omega
ω
at
A
A
A
intersects
B
C
BC
BC
at
P
P
P
,
B
B
B
lies between
P
P
P
and
C
C
C
. Let
M
M
M
be the midpoint of the minor arc
A
B
AB
A
B
.
M
P
MP
MP
intersects
ω
\omega
ω
at
Q
Q
Q
. Point
X
X
X
lies on a ray
P
A
PA
P
A
such that
∠
X
C
B
=
90
\angle XCB=90
∠
XCB
=
90
. Prove that line
X
Q
XQ
XQ
passes through the orthocenter of the triangle
A
B
O
ABO
A
BO
Mayya Golitsyna
geometry