MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2016 China Second Round Olympiad
2
2
Part of
2016 China Second Round Olympiad
Problems
(1)
Geometry
Source: 2016 China Second Round Olympiad Problem 2
1/8/2017
Let
X
,
Y
X,Y
X
,
Y
be two points which lies on the line
B
C
BC
BC
of
△
A
B
C
(
X
,
B
,
C
,
Y
lies in sequence
)
\triangle ABC(X,B,C,Y\text{lies in sequence})
△
A
BC
(
X
,
B
,
C
,
Y
lies in sequence
)
such that
B
X
⋅
A
C
=
C
Y
⋅
A
B
BX\cdot AC=CY\cdot AB
BX
⋅
A
C
=
C
Y
⋅
A
B
,
O
1
,
O
2
O_1,O_2
O
1
,
O
2
are the circumcenters of
△
A
C
X
,
△
A
B
Y
\triangle ACX,\triangle ABY
△
A
CX
,
△
A
B
Y
,
O
1
O
2
∩
A
B
=
U
,
O
1
O
2
∩
A
C
=
V
O_1O_2\cap AB=U,O_1O_2\cap AC=V
O
1
O
2
∩
A
B
=
U
,
O
1
O
2
∩
A
C
=
V
. Prove that
△
A
U
V
\triangle AUV
△
A
U
V
is a isosceles triangle.
geometry