MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgarian Winter Tournament
2024 Bulgarian Winter Tournament
12.2
Spiral similarity geo
Spiral similarity geo
Source: Bulgarian Winter Tournament 2024 12.2
January 28, 2024
geometry
Problem Statement
Let
A
B
C
ABC
A
BC
be scalene and acute triangle with
C
A
>
C
B
CA>CB
C
A
>
CB
and let
P
P
P
be an internal point, satisfying
∠
A
P
B
=
18
0
∘
−
∠
A
C
B
\angle APB=180^{\circ}-\angle ACB
∠
A
PB
=
18
0
∘
−
∠
A
CB
; the lines
A
P
,
B
P
AP, BP
A
P
,
BP
meet
B
C
,
C
A
BC, CA
BC
,
C
A
at
A
1
,
B
1
A_1, B_1
A
1
,
B
1
. If
M
M
M
is the midpoint of
A
1
B
1
A_1B_1
A
1
B
1
and
(
A
1
B
1
C
)
(A_1B_1C)
(
A
1
B
1
C
)
meets
(
A
B
C
)
(ABC)
(
A
BC
)
at
Q
Q
Q
, show that
∠
P
Q
M
=
∠
B
Q
A
1
\angle PQM=\angle BQA_1
∠
PQM
=
∠
BQ
A
1
.
Back to Problems
View on AoPS