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2018 JBMO Shortlist
G1
G1
Part of
2018 JBMO Shortlist
Problems
(1)
MN bisects segment CH
Source: JBMO Shortlist 2018 G1
7/22/2019
Let
H
H
H
be the orthocentre of an acute triangle
A
B
C
ABC
A
BC
with
B
C
>
A
C
BC > AC
BC
>
A
C
, inscribed in a circle
Γ
\Gamma
Γ
. The circle with centre
C
C
C
and radius
C
B
CB
CB
intersects
Γ
\Gamma
Γ
at the point
D
D
D
, which is on the arc
A
B
AB
A
B
not containing
C
C
C
. The circle with centre
C
C
C
and radius
C
A
CA
C
A
intersects the segment
C
D
CD
C
D
at the point
K
K
K
. The line parallel to
B
D
BD
B
D
through
K
K
K
, intersects
A
B
AB
A
B
at point
L
L
L
. If
M
M
M
is the midpoint of
A
B
AB
A
B
and
N
N
N
is the foot of the perpendicular from
H
H
H
to
C
L
CL
C
L
, prove that the line
M
N
MN
MN
bisects the segment
C
H
CH
C
H
.
geometry
bisects segment