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APMO
2018 APMO
1
1
Part of
2018 APMO
Problems
(1)
Circles tangent at orthocenter
Source: APMO 2018 P1
6/24/2018
Let
H
H
H
be the orthocenter of the triangle
A
B
C
ABC
A
BC
. Let
M
M
M
and
N
N
N
be the midpoints of the sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Assume that
H
H
H
lies inside the quadrilateral
B
M
N
C
BMNC
BMNC
and that the circumcircles of triangles
B
M
H
BMH
BM
H
and
C
N
H
CNH
CN
H
are tangent to each other. The line through
H
H
H
parallel to
B
C
BC
BC
intersects the circumcircles of the triangles
B
M
H
BMH
BM
H
and
C
N
H
CNH
CN
H
in the points
K
K
K
and
L
L
L
, respectively. Let
F
F
F
be the intersection point of
M
K
MK
M
K
and
N
L
NL
N
L
and let
J
J
J
be the incenter of triangle
M
H
N
MHN
M
H
N
. Prove that
F
J
=
F
A
F J = F A
F
J
=
F
A
.
geometry
APMO