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Balkan MO Shortlist
2010 Balkan MO Shortlist
G6
G6
Part of
2010 Balkan MO Shortlist
Problems
(1)
circumcircle k of ABC passes through P iff k passes through midpoint of EF
Source: 2010 Balkan Shortlist G6 BMO
4/4/2020
In a triangle
A
B
C
ABC
A
BC
the excircle at the side
B
C
BC
BC
touches
B
C
BC
BC
in point
D
D
D
and the lines
A
B
AB
A
B
and
A
C
AC
A
C
in points
E
E
E
and
F
F
F
respectively. Let
P
P
P
be the projection of
D
D
D
on
E
F
EF
EF
. Prove that the circumcircle
k
k
k
of the triangle
A
B
C
ABC
A
BC
passes through
P
P
P
if and only if
k
k
k
passes through the midpoint
M
M
M
of the segment
E
F
EF
EF
.
geometry
circumcircle