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National and Regional Contests
Thailand Contests
Thailand Online MO
2022 Thailand Online MO
8
8
Part of
2022 Thailand Online MO
Problems
(1)
Prove that two circumcircle intersect on a line
Source: 2022 Thailand Online MO P8
4/4/2022
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
A
D
=
B
C
AD = BC
A
D
=
BC
,
∠
B
A
C
+
∠
D
C
A
=
18
0
∘
\angle BAC+\angle DCA = 180^{\circ}
∠
B
A
C
+
∠
D
C
A
=
18
0
∘
, and
∠
B
A
C
≠
9
0
∘
.
\angle BAC \neq 90^{\circ}.
∠
B
A
C
=
9
0
∘
.
Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the circumcenters of triangles
A
B
C
ABC
A
BC
and
C
A
D
CAD
C
A
D
, respectively. Prove that one intersection point of the circumcircles of triangles
O
1
B
C
O_1BC
O
1
BC
and
O
2
A
D
O_2AD
O
2
A
D
lies on
A
C
AC
A
C
.
geometry
circumcircle