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Problem 2, Geometry

Source: Silk Road Mathematical Competition 2017, P2

May 25, 2017

geometry proposedgeometry

Problem Statement

The quadrilateral

ABCD

is inscribed in the circle ω. The diagonals

AC

and

E_1

intersect at the point

O

. On the segments

AO

and

DO

, the points

E

and

F

are chosen, respectively. The straight line

EF

intersects ω at the points

E_1

and

F_1

. The circumscribed circles of the triangles

ADE

and

BCF

intersect the segment

EF

at the points

E_2

and

F_2

respectively (assume that all the points

E, F, E_1, F_1, E_2

and

F_2

are different). Prove that

E_1E_2 = F_1F_2

.

(N. Sedrakyan)

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