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National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2015 JBMO TST - Turkey
2
2
Part of
2015 JBMO TST - Turkey
Problems
(1)
Tangencies in Quadrilateral
Source: Turkey JBMO TST 2015 P2
6/23/2016
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and let
ω
\omega
ω
be a circle tangent to the lines
A
B
AB
A
B
and
B
C
BC
BC
at points
A
A
A
and
C
C
C
, respectively.
ω
\omega
ω
intersects the line segments
A
D
AD
A
D
and
C
D
CD
C
D
again at
E
E
E
and
F
F
F
, respectively, which are both different from
D
D
D
. Let
G
G
G
be the point of intersection of the lines
A
F
AF
A
F
and
C
E
CE
CE
. Given
∠
A
C
B
=
∠
G
D
C
+
∠
A
C
E
\angle ACB=\angle GDC+\angle ACE
∠
A
CB
=
∠
G
D
C
+
∠
A
CE
, prove that the line
A
D
AD
A
D
is tangent to th circumcircle of the triangle
A
G
B
AGB
A
GB
.
geometry