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2018 Benelux
3
3
Part of
2018 Benelux
Problems
(1)
Benelux Olympiad 2018, Problem 3
Source: BxMO 2018, Problem 3
4/28/2018
Let
A
B
C
ABC
A
BC
be a triangle with orthocentre
H
H
H
, and let
D
D
D
,
E
E
E
, and
F
F
F
denote the respective midpoints of line segments
A
B
AB
A
B
,
A
C
AC
A
C
, and
A
H
AH
A
H
. The reflections of
B
B
B
and
C
C
C
in
F
F
F
are
P
P
P
and
Q
Q
Q
, respectively. (a) Show that lines
P
E
PE
PE
and
Q
D
QD
Q
D
intersect on the circumcircle of triangle
A
B
C
ABC
A
BC
. (b) Prove that lines
P
D
PD
P
D
and
Q
E
QE
QE
intersect on line segment
A
H
AH
A
H
.
BxMO
Benelux
geometry