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IMO Shortlist
2015 IMO Shortlist
G7
G7
Part of
2015 IMO Shortlist
Problems
(1)
four incircles
Source: 2015 IMO Shortlist G7
7/7/2016
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, and let
P
P
P
,
Q
Q
Q
,
R
R
R
, and
S
S
S
be points on the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
, and
D
A
DA
D
A
, respectively. Let the line segment
P
R
PR
PR
and
Q
S
QS
QS
meet at
O
O
O
. Suppose that each of the quadrilaterals
A
P
O
S
APOS
A
POS
,
B
Q
O
P
BQOP
BQOP
,
C
R
O
Q
CROQ
CROQ
, and
D
S
O
R
DSOR
D
SOR
has an incircle. Prove that the lines
A
C
AC
A
C
,
P
Q
PQ
PQ
, and
R
S
RS
RS
are either concurrent or parallel to each other.
IMO Shortlist
geometry