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Show that a quadrilateral is a square (2015 OMCS #4)

Source:

May 16, 2015

geometryQuadrilaterals

Problem Statement

Let

ABCD

be a convex quadrilateral such that

\angle{BAD} = 90^{\circ}

and its diagonals

AC

and

BD

are perpendicular. Let

M

be the midpoint of side

CD

, and

E

be the intersection of

BM

and

AC

. Let

F

be a point on side

AD

such that

BM

and

EF

are perpendicular. If

CE = AF\sqrt{2}

and

FD = CE\sqrt{2}

, show that

ABCD

is a square.

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