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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey EGMO TST
2014 Turkey EGMO TST
5
5
Part of
2014 Turkey EGMO TST
Problems
(1)
Circles Tangent to Sides and Circumcircle
Source: Turkey EGMO TST 2014 P5
8/6/2016
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
ω
\omega
ω
and let
ω
A
\omega_A
ω
A
be a circle drawn outside
A
B
C
ABC
A
BC
and tangent to side
B
C
BC
BC
at
A
1
A_1
A
1
and tangent to
ω
\omega
ω
at
A
2
A_2
A
2
. Let the circles
ω
B
\omega_B
ω
B
and
ω
C
\omega_C
ω
C
and the points
B
1
,
B
2
,
C
1
,
C
2
B_1, B_2, C_1, C_2
B
1
,
B
2
,
C
1
,
C
2
are defined similarly. Prove that if the lines
A
A
1
,
B
B
1
,
C
C
1
AA_1, BB_1, CC_1
A
A
1
,
B
B
1
,
C
C
1
are concurrent, then the lines
A
A
2
,
B
B
2
,
C
C
2
AA_2, BB_2, CC_2
A
A
2
,
B
B
2
,
C
C
2
are also concurrent.
geometry
circumcircle