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Kyiv City MO
Kyiv City MO - geometry
Kyiv City MO Seniors Round2 2010+ geometry
2021.10.4
2021.10.4
Part of
Kyiv City MO Seniors Round2 2010+ geometry
Problems
(1)
concyclic, <OAD+<OBC= <ODA + <OCB = 90^o (2021 Kyiv City MO Round2 10.4)
Source:
2/14/2021
Inside the quadrilateral
A
B
C
D
ABCD
A
BC
D
marked a point
O
O
O
such that
∠
O
A
D
+
∠
O
B
C
=
∠
O
D
A
+
∠
O
C
B
=
9
0
o
\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o
∠
O
A
D
+
∠
OBC
=
∠
O
D
A
+
∠
OCB
=
9
0
o
. Prove that the centers of the circumscribed circles around triangles
O
A
D
OAD
O
A
D
and
O
B
C
OBC
OBC
as well as the midpoints of the sides
A
B
AB
A
B
and
C
D
CD
C
D
lie on one circle.(Anton Trygub)
geometry
Concyclic
angles