MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2008 Korea Junior Math Olympiad
1
collinear points given product of 4 ratios =1
collinear points given product of 4 ratios =1
Source: KJMO 2008 p1
May 2, 2019
ratio
geometry
Concyclic
collinear
Problem Statement
In a
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
, points
A
,
B
A,B
A
,
B
lie on segment
Z
X
,
C
,
D
ZX, C,D
ZX
,
C
,
D
lie on segment
X
Y
,
E
,
F
XY , E, F
X
Y
,
E
,
F
lie on segment
Y
Z
YZ
Y
Z
.
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
lie on a circle, and
A
Z
⋅
E
Y
⋅
Z
B
⋅
Y
F
E
Z
⋅
C
Y
⋅
Z
F
⋅
Y
D
=
1
\frac{AZ \cdot EY \cdot ZB \cdot Y F}{EZ \cdot CY \cdot ZF \cdot Y D}= 1
EZ
⋅
C
Y
⋅
ZF
⋅
Y
D
A
Z
⋅
E
Y
⋅
ZB
⋅
Y
F
=
1
. Let
L
=
Z
X
∩
D
E
L = ZX \cap DE
L
=
ZX
∩
D
E
,
M
=
X
Y
∩
A
F
M = XY \cap AF
M
=
X
Y
∩
A
F
,
N
=
Y
Z
∩
B
C
N = Y Z \cap BC
N
=
Y
Z
∩
BC
. Prove that
L
,
M
,
N
L,M,N
L
,
M
,
N
are collinear.
Back to Problems
View on AoPS