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International Contests
Romanian Masters of Mathematics Collection
2016 Romanian Master of Mathematics
1
1
Part of
2016 Romanian Master of Mathematics
Problems
(1)
Concurrent lines(or parallel)
Source: RMM 2016 Day 1 Problem 1
2/27/2016
Let
A
B
C
ABC
A
BC
be a triangle and let
D
D
D
be a point on the segment
B
C
,
D
≠
B
BC, D\neq B
BC
,
D
=
B
and
D
≠
C
D\neq C
D
=
C
. The circle
A
B
D
ABD
A
B
D
meets the segment
A
C
AC
A
C
again at an interior point
E
E
E
. The circle
A
C
D
ACD
A
C
D
meets the segment
A
B
AB
A
B
again at an interior point
F
F
F
. Let
A
′
A'
A
′
be the reflection of
A
A
A
in the line
B
C
BC
BC
. The lines
A
′
C
A'C
A
′
C
and
D
E
DE
D
E
meet at
P
P
P
, and the lines
A
′
B
A'B
A
′
B
and
D
F
DF
D
F
meet at
Q
Q
Q
. Prove that the lines
A
D
,
B
P
AD, BP
A
D
,
BP
and
C
Q
CQ
CQ
are concurrent (or all parallel).
geometry
RMM
RMM 2016