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Cono Sur Olympiad
2015 Cono Sur Olympiad
4
4
Part of
2015 Cono Sur Olympiad
Problems
(1)
Show that a quadrilateral is a square (2015 OMCS #4)
Source:
5/16/2015
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
∠
B
A
D
=
9
0
∘
\angle{BAD} = 90^{\circ}
∠
B
A
D
=
9
0
∘
and its diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular. Let
M
M
M
be the midpoint of side
C
D
CD
C
D
, and
E
E
E
be the intersection of
B
M
BM
BM
and
A
C
AC
A
C
. Let
F
F
F
be a point on side
A
D
AD
A
D
such that
B
M
BM
BM
and
E
F
EF
EF
are perpendicular. If
C
E
=
A
F
2
CE = AF\sqrt{2}
CE
=
A
F
2
and
F
D
=
C
E
2
FD = CE\sqrt{2}
F
D
=
CE
2
, show that
A
B
C
D
ABCD
A
BC
D
is a square.
geometry
Quadrilaterals