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Thailand Contests
Thailand National Olympiad
2022 Thailand Mathematical Olympiad
1
1
Part of
2022 Thailand Mathematical Olympiad
Problems
(1)
Equal segment wanted
Source: 2022 Thailand MO Day 1 P1
6/15/2022
Let
B
C
BC
BC
be a chord of a circle
Γ
\Gamma
Γ
and
A
A
A
be a point inside
Γ
\Gamma
Γ
such that
∠
B
A
C
\angle BAC
∠
B
A
C
is acute. Outside
△
A
B
C
\triangle ABC
△
A
BC
, construct two isosceles triangles
△
A
C
P
\triangle ACP
△
A
CP
and
△
A
B
R
\triangle ABR
△
A
BR
such that
∠
A
C
P
\angle ACP
∠
A
CP
and
∠
A
B
R
\angle ABR
∠
A
BR
are right angles. Let lines
B
A
BA
B
A
and
C
A
CA
C
A
meet
Γ
\Gamma
Γ
again at points
E
E
E
and
F
F
F
, respectively. Let lines
E
P
EP
EP
and
F
R
FR
FR
meet
Γ
\Gamma
Γ
again at points
X
X
X
and
Y
Y
Y
, respectively. Prove that
B
X
=
C
Y
BX=CY
BX
=
C
Y
.
geometry