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Problems
Contests
National and Regional Contests
Mexico Contests
Mexican Quarantine Mathematical Olympiad
#3
#3
Part of
Mexican Quarantine Mathematical Olympiad
Problems
(1)
Two circles and concyclic points
Source: Mexican Quarantine Mathematical Olympiad P3
4/25/2020
Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be circles intersecting at points
A
A
A
and
B
B
B
. A line through
A
A
A
intersects
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
at
C
C
C
and
D
D
D
respectively. Let
P
P
P
be the intersection of the lines tangent to
Γ
1
\Gamma_1
Γ
1
at
A
A
A
and
C
C
C
, and let
Q
Q
Q
be the intersection of the lines tangent to
Γ
2
\Gamma_2
Γ
2
at
A
A
A
and
D
D
D
. Let
X
X
X
be the second intersection point of the circumcircles of
B
C
P
BCP
BCP
and
B
D
Q
BDQ
B
D
Q
, and let
Y
Y
Y
be the intersection of lines
A
B
AB
A
B
and
P
Q
PQ
PQ
. Prove that
C
C
C
,
D
D
D
,
X
X
X
and
Y
Y
Y
are concyclic.Proposed by Ariel García
geometry
Concyclic
tangent