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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2022 Sharygin Geometry Olympiad
16
16
Part of
2022 Sharygin Geometry Olympiad
Problems
(1)
Feet of angle bisectors and two vertices are Concyclic!!
Source: Sharygin 2022 P16
3/4/2022
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral,
E
=
A
C
∩
B
D
E = AC \cap BD
E
=
A
C
∩
B
D
,
F
=
A
D
∩
B
C
F = AD \cap BC
F
=
A
D
∩
BC
. The bisectors of angles
A
F
B
AFB
A
FB
and
A
E
B
AEB
A
EB
meet
C
D
CD
C
D
at points
X
,
Y
X, Y
X
,
Y
. Prove that
A
,
B
,
X
,
Y
A, B, X, Y
A
,
B
,
X
,
Y
are concyclic.
geometry
angle bisector