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Problems
Contests
International Contests
IberoAmerican
2016 IberoAmerican
5
5
Part of
2016 IberoAmerican
Problems
(1)
A, E and F are collinear
Source: Iberoamerican Olympiad 2016-P5
9/28/2016
The circumferences
C
1
C_1
C
1
and
C
2
C_2
C
2
cut each other at different points
A
A
A
and
K
K
K
. The common tangent to
C
1
C_1
C
1
and
C
2
C_2
C
2
nearer to
K
K
K
touches
C
1
C_1
C
1
at
B
B
B
and
C
2
C_2
C
2
at
C
C
C
. Let
P
P
P
be the foot of the perpendicular from
B
B
B
to
A
C
AC
A
C
, and let
Q
Q
Q
be the foot of the perpendicular from
C
C
C
to
A
B
AB
A
B
. If
E
E
E
and
F
F
F
are the symmetric points of
K
K
K
with respect to the lines
P
Q
PQ
PQ
and
B
C
BC
BC
, respectively, prove that
A
,
E
A, E
A
,
E
and
F
F
F
are collinear.
geometry