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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2023 Moldova Team Selection Test
10
10
Part of
2023 Moldova Team Selection Test
Problems
(1)
Prove that line $QM$ passes through the midpoint of $AC.$
Source: Moldova TST 2023
4/9/2023
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
C
B
=
90
\angle ACB=90
∠
A
CB
=
90
and
A
C
>
B
C
.
AC>BC.
A
C
>
BC
.
Let
Ω
\Omega
Ω
be the circumcircle of
A
B
C
.
ABC.
A
BC
.
Point
D
D
D
is the midpoint of small arc
A
C
AC
A
C
of
Ω
.
\Omega.
Ω.
Point
M
M
M
is symmetric with
A
A
A
with respect to
D
.
D.
D
.
Point
N
N
N
is the midpoint of
M
C
.
MC.
MC
.
Line
A
N
AN
A
N
intersects
Ω
\Omega
Ω
in point
P
P
P
and line
B
P
BP
BP
intersects line
D
N
DN
D
N
in point
Q
.
Q.
Q
.
Prove that line
Q
M
QM
QM
passes through the midpoint of
A
C
.
AC.
A
C
.
geometry