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National and Regional Contests
Azerbaijan Contests
Azerbaijan Team Selection Test
2016 Azerbaijan Team Selection Test
1
1
Part of
2016 Azerbaijan Team Selection Test
Problems
(1)
Tangent Circles
Source: Azerbaijan IMO TST 2016,D1 P1
5/26/2018
Tangents from the point
A
A
A
to the circle
Γ
\Gamma
Γ
touche this circle at
C
C
C
and
D
D
D
.Let
B
B
B
be a point on
Γ
\Gamma
Γ
,different from
C
C
C
and
D
D
D
. The circle
ω
\omega
ω
that passes through points
A
A
A
and
B
B
B
intersect with lines
A
C
AC
A
C
and
A
D
AD
A
D
at
F
F
F
and
E
E
E
,respectively.Prove that the circumcircles of triangles
A
B
C
ABC
A
BC
and
D
E
B
DEB
D
EB
are tangent if and only if the points
C
,
D
,
F
C,D,F
C
,
D
,
F
and
E
E
E
are cyclic.
geometry
circumcircle