MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2024 Singapore MO Open
Q1
Q1
Part of
2024 Singapore MO Open
Problems
(1)
Easy right-angled triangle problem
Source: SMO open 2024 Q1
8/3/2024
In triangle
A
B
C
ABC
A
BC
,
∠
B
=
9
0
∘
\angle B=90^\circ
∠
B
=
9
0
∘
,
A
B
>
B
C
AB>BC
A
B
>
BC
, and
P
P
P
is the point such that
B
P
=
B
C
BP=BC
BP
=
BC
and
∠
A
P
B
=
9
0
∘
\angle APB=90^\circ
∠
A
PB
=
9
0
∘
, where
P
P
P
and
C
C
C
lie on the same side of
A
B
AB
A
B
. Let
Q
Q
Q
be the point on
A
B
AB
A
B
such that
A
P
=
A
Q
AP=AQ
A
P
=
A
Q
, and let
M
M
M
be the midpoint of
Q
C
QC
QC
. Prove that the line through
M
M
M
parallel to
A
P
AP
A
P
passes through the midpoint of
A
B
AB
A
B
.
geometry