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National and Regional Contests
Thailand Contests
Thailand Online MO
2022 Thailand Online MO
2
2
Part of
2022 Thailand Online MO
Problems
(1)
Prove that intersection of two circumcircle lie on a line
Source: 2022 Thailand Online MO P2
4/4/2022
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid such that
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
and
A
B
>
C
D
AB > CD
A
B
>
C
D
. Points
X
X
X
and
Y
Y
Y
are on the side
A
B
AB
A
B
such that
X
Y
=
A
B
−
C
D
XY = AB-CD
X
Y
=
A
B
−
C
D
and
X
X
X
lies between
A
A
A
and
Y
Y
Y
. Prove that one intersection of the circumcircles of triangles
A
Y
D
AYD
A
Y
D
and
B
X
C
BXC
BXC
is on line
C
D
CD
C
D
.
geometry
circumcircle