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Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2021 Ukraine National Mathematical Olympiad
8.4
8.4
Part of
2021 Ukraine National Mathematical Olympiad
Problems
(1)
Two points inside isosceles triangle
Source: Ukraine MO 2021 8.4
5/2/2021
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
. Points
P
P
P
and
Q
Q
Q
inside
△
A
B
C
\triangle ABC
△
A
BC
are such that
∠
B
P
C
=
3
2
∠
B
A
C
\angle BPC = \frac{3}{2} \angle BAC
∠
BPC
=
2
3
∠
B
A
C
,
B
P
=
A
Q
BP = AQ
BP
=
A
Q
and
A
P
=
C
Q
AP = CQ
A
P
=
CQ
. Prove that
A
P
=
P
Q
AP = PQ
A
P
=
PQ
.Proposed by Fedir Yudin
geometry