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Vectorial geometry relating concyclic quadrilateral

Source: Romanian National Olympiad 2014, Grade IX, Problem 4

March 2, 2019

geometry

Problem Statement

Let

ABCD

be a quadrilateral inscribed in a circle of diameter

AC.

Fix points

E,F

of segments

CD,

respectively,

BC

such that

AE

is perpendicular to

DF

and

AF

is perpendicular to

BE.

Show that

AB=AD.