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Contests
National and Regional Contests
South Africa Contests
South Africa National Olympiad
2015 South Africa National Olympiad
4
4
Part of
2015 South Africa National Olympiad
Problems
(1)
Let $ABC$ be an acute-angled triangle with $AB < AC$ ...
Source:
8/8/2015
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
, and let points
D
D
D
and
E
E
E
be chosen on the side
A
C
AC
A
C
and
B
C
BC
BC
respectively in such a way that
A
D
=
A
E
=
A
B
AD = AE = AB
A
D
=
A
E
=
A
B
. The circumcircle of
A
B
E
ABE
A
BE
intersects the line
A
C
AC
A
C
at
A
A
A
and
F
F
F
and the line
D
E
DE
D
E
at
E
E
E
and
P
P
P
. Prove that
P
P
P
is the circumcentre of
B
D
F
BDF
B
D
F
.
geometry
circumcircle