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Mediterranean Mathematics Olympiad
2016 Mediterranean Mathematics Olympiad
1
1
Part of
2016 Mediterranean Mathematics Olympiad
Problems
(1)
Orthogonal circumradius
Source: Mediterranean Mathematics Olympiad 2016 P1
6/4/2016
Let
A
B
C
ABC
A
BC
be a triangle. Let
D
D
D
be the intersection point of the angle bisector at
A
A
A
with
B
C
BC
BC
. Let
T
T
T
be the intersection point of the tangent line to the circumcircle of triangle
A
B
C
ABC
A
BC
at point
A
A
A
with the line through
B
B
B
and
C
C
C
. Let
I
I
I
be the intersection point of the orthogonal line to
A
T
AT
A
T
through point
D
D
D
with the altitude
h
a
h_a
h
a
of the triangle at point
A
A
A
. Let
P
P
P
be the midpoint of
A
B
AB
A
B
, and let
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
. Let
M
M
M
be the intersection point of
A
B
AB
A
B
and
T
I
TI
T
I
, and let
F
F
F
be the intersection point of
P
T
PT
PT
and
A
D
AD
A
D
. Prove:
M
F
MF
MF
and
A
O
AO
A
O
are orthogonal to each other.
geometry
circumcircle