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APMO
2014 APMO
5
5
Part of
2014 APMO
Problems
(1)
Show that a circumcircle is tangent to Omega
Source: APMO 2014 Problem 5
3/28/2014
Circles
ω
\omega
ω
and
Ω
\Omega
Ω
meet at points
A
A
A
and
B
B
B
. Let
M
M
M
be the midpoint of the arc
A
B
AB
A
B
of circle
ω
\omega
ω
(
M
M
M
lies inside
Ω
\Omega
Ω
). A chord
M
P
MP
MP
of circle
ω
\omega
ω
intersects
Ω
\Omega
Ω
at
Q
Q
Q
(
Q
Q
Q
lies inside
ω
\omega
ω
). Let
ℓ
P
\ell_P
ℓ
P
be the tangent line to
ω
\omega
ω
at
P
P
P
, and let
ℓ
Q
\ell_Q
ℓ
Q
be the tangent line to
Ω
\Omega
Ω
at
Q
Q
Q
. Prove that the circumcircle of the triangle formed by the lines
ℓ
P
\ell_P
ℓ
P
,
ℓ
Q
\ell_Q
ℓ
Q
and
A
B
AB
A
B
is tangent to
Ω
\Omega
Ω
.Ilya Bogdanov, Russia and Medeubek Kungozhin, Kazakhstan
geometry
circumcircle
APMO