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National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
2017 Turkey MO (2nd round)
2
2
Part of
2017 Turkey MO (2nd round)
Problems
(1)
Turkey NMO 2017 p2
Source:
1/25/2018
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral such that line
A
B
AB
A
B
intersects
C
D
CD
C
D
at
X
X
X
. Denote circles with inradius
r
1
r_1
r
1
and centers
A
,
B
A, B
A
,
B
as
w
a
w_a
w
a
and
w
b
w_b
w
b
with inradius
r
2
r_2
r
2
and centers
C
,
D
C, D
C
,
D
as
w
c
w_c
w
c
and
w
d
w_d
w
d
.
w
a
w_a
w
a
intersects
w
d
w_d
w
d
at
P
,
Q
P, Q
P
,
Q
.
w
b
w_b
w
b
intersects
w
c
w_c
w
c
at
R
,
S
R, S
R
,
S
. Prove that if
X
A
.
X
B
+
r
2
2
=
X
C
.
X
D
+
r
1
2
XA.XB+r_2^2=XC.XD+r_1^2
X
A
.
XB
+
r
2
2
=
XC
.
X
D
+
r
1
2
, then
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
are cyclic.
geometry