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Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2015 China Western Mathematical Olympiad
2
2
Part of
2015 China Western Mathematical Olympiad
Problems
(1)
A property of touching circles
Source: CWMO P2
8/20/2015
Two circles
(
Ω
1
)
,
(
Ω
2
)
\left(\Omega_1\right),\left(\Omega_2\right)
(
Ω
1
)
,
(
Ω
2
)
touch internally on the point
T
T
T
. Let
M
,
N
M,N
M
,
N
be two points on the circle
(
Ω
1
)
\left(\Omega_1\right)
(
Ω
1
)
which are different from
T
T
T
and
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be four points on
(
Ω
2
)
\left(\Omega_2\right)
(
Ω
2
)
such that the chords
A
B
,
C
D
AB, CD
A
B
,
C
D
pass through
M
,
N
M,N
M
,
N
, respectively. Prove that if
A
C
,
B
D
,
M
N
AC,BD,MN
A
C
,
B
D
,
MN
have a common point
K
K
K
, then
T
K
TK
T
K
is the angle bisector of
∠
M
T
N
\angle MTN
∠
MTN
.*
(
Ω
2
)
\left(\Omega_2\right)
(
Ω
2
)
is bigger than
(
Ω
1
)
\left(\Omega_1\right)
(
Ω
1
)
geometry
angle bisector