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Balkan MO Shortlist
2023 Balkan MO Shortlist
G1
G1
Part of
2023 Balkan MO Shortlist
Problems
(1)
Circumscribed quadrilateral and incenters
Source: BMO SL 2023 G1
5/3/2024
Let
A
B
C
D
ABCD
A
BC
D
be a circumscribed quadrilateral and let
X
X
X
be the intersection point of its diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Let
I
1
,
I
2
,
I
3
,
I
4
I_1, I_2, I_3, I_4
I
1
,
I
2
,
I
3
,
I
4
be the incenters of
△
D
X
C
\triangle DXC
△
D
XC
,
△
B
X
C
\triangle BXC
△
BXC
,
△
A
X
B
\triangle AXB
△
A
XB
, and
△
D
X
A
\triangle DXA
△
D
X
A
, respectively. The circumcircle of
△
C
I
1
I
2
\triangle CI_1I_2
△
C
I
1
I
2
intersects the sides
C
B
CB
CB
and
C
D
CD
C
D
at points
P
P
P
and
Q
Q
Q
, respectively. The circumcircle of
△
A
I
3
I
4
\triangle AI_3I_4
△
A
I
3
I
4
intersects the sides
A
B
AB
A
B
and
A
D
AD
A
D
at points
M
M
M
and
N
N
N
, respectively. Prove that
A
M
+
C
Q
=
A
N
+
C
P
AM+CQ=AN+CP
A
M
+
CQ
=
A
N
+
CP
geometry
incenter