MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2013 India National Olympiad
1
1
Part of
2013 India National Olympiad
Problems
(1)
O_1P, O_2Q tangents to (O_1), (O_2) [INMO 2013 P1]
Source:
2/3/2013
Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be two circles touching each other externally at
R
.
R.
R
.
Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the centres of
Γ
1
\Gamma_1
Γ
1
and
Γ
2
,
\Gamma_2,
Γ
2
,
respectively. Let
ℓ
1
\ell_1
ℓ
1
be a line which is tangent to
Γ
2
\Gamma_2
Γ
2
at
P
P
P
and passing through
O
1
,
O_1,
O
1
,
and let
ℓ
2
\ell_2
ℓ
2
be the line tangent to
Γ
1
\Gamma_1
Γ
1
at
Q
Q
Q
and passing through
O
2
.
O_2.
O
2
.
Let
K
=
ℓ
1
∩
ℓ
2
.
K=\ell_1\cap \ell_2.
K
=
ℓ
1
∩
ℓ
2
.
If
K
P
=
K
Q
KP=KQ
K
P
=
K
Q
then prove that the triangle
P
Q
R
PQR
PQR
is equilateral.
geometry
circumcircle