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Philippine MO
2011 Philippine MO
2
2
Part of
2011 Philippine MO
Problems
(1)
Perpendicularity from Midpoints
Source: 2011 Philippine Math Olympiad National Stage Problem 2
7/15/2013
In triangle
A
B
C
ABC
A
BC
, let
X
X
X
and
Y
Y
Y
be the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. On segment
B
C
BC
BC
, there is a point
D
D
D
, different from its midpoint, such that
∠
X
D
Y
=
∠
B
A
C
\angle{XDY}=\angle{BAC}
∠
X
D
Y
=
∠
B
A
C
. Prove that
A
D
⊥
B
C
AD\perp BC
A
D
⊥
BC
.
geometry
geometric transformation
reflection