MathDB
Problems
Contests
International Contests
IMO Shortlist
2022 IMO Shortlist
G8
G8
Part of
2022 IMO Shortlist
Problems
(1)
Cyclic hexagon with incircles
Source: ISL 2022 G8
7/9/2023
Let
A
A
′
B
C
C
′
B
′
AA'BCC'B'
A
A
′
BC
C
′
B
′
be a convex cyclic hexagon such that
A
C
AC
A
C
is tangent to the incircle of the triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
, and
A
′
C
′
A'C'
A
′
C
′
is tangent to the incircle of the triangle
A
B
C
ABC
A
BC
. Let the lines
A
B
AB
A
B
and
A
′
B
′
A'B'
A
′
B
′
meet at
X
X
X
and let the lines
B
C
BC
BC
and
B
′
C
′
B'C'
B
′
C
′
meet at
Y
Y
Y
.Prove that if
X
B
Y
B
′
XBYB'
XB
Y
B
′
is a convex quadrilateral, then it has an incircle.
geometry