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Contests
International Contests
IMO Shortlist
2015 IMO Shortlist
G5
G5
Part of
2015 IMO Shortlist
Problems
(1)
Two lengths are equal
Source: IMO 2015 Shortlist, G5
7/7/2016
Let
A
B
C
ABC
A
BC
be a triangle with
C
A
≠
C
B
CA \neq CB
C
A
=
CB
. Let
D
D
D
,
F
F
F
, and
G
G
G
be the midpoints of the sides
A
B
AB
A
B
,
A
C
AC
A
C
, and
B
C
BC
BC
respectively. A circle
Γ
\Gamma
Γ
passing through
C
C
C
and tangent to
A
B
AB
A
B
at
D
D
D
meets the segments
A
F
AF
A
F
and
B
G
BG
BG
at
H
H
H
and
I
I
I
, respectively. The points
H
′
H'
H
′
and
I
′
I'
I
′
are symmetric to
H
H
H
and
I
I
I
about
F
F
F
and
G
G
G
, respectively. The line
H
′
I
′
H'I'
H
′
I
′
meets
C
D
CD
C
D
and
F
G
FG
FG
at
Q
Q
Q
and
M
M
M
, respectively. The line
C
M
CM
CM
meets
Γ
\Gamma
Γ
again at
P
P
P
. Prove that
C
Q
=
Q
P
CQ = QP
CQ
=
QP
.Proposed by El Salvador
geometry