MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2015 China National Olympiad
2
Two ratio are equal
Two ratio are equal
Source: 2015 China National Olympiad
December 20, 2014
ratio
trigonometry
geometry
Problem Statement
Let
A
,
B
,
D
,
E
,
F
,
C
A, B, D, E, F, C
A
,
B
,
D
,
E
,
F
,
C
be six points lie on a circle (in order) satisfy
A
B
=
A
C
AB=AC
A
B
=
A
C
. Let
P
=
A
D
∩
B
E
,
R
=
A
F
∩
C
E
,
Q
=
B
F
∩
C
D
,
S
=
A
D
∩
B
F
,
T
=
A
F
∩
C
D
P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD
P
=
A
D
∩
BE
,
R
=
A
F
∩
CE
,
Q
=
BF
∩
C
D
,
S
=
A
D
∩
BF
,
T
=
A
F
∩
C
D
. Let
K
K
K
be a point lie on
S
T
ST
ST
satisfy
∠
Q
K
S
=
∠
E
C
A
\angle QKS=\angle ECA
∠
Q
K
S
=
∠
EC
A
.Prove that
S
K
K
T
=
P
Q
Q
R
\frac{SK}{KT}=\frac{PQ}{QR}
K
T
S
K
=
QR
PQ
Back to Problems
View on AoPS