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Romania TST 2016 Day 1 P1

Source: Romania TST 2016 Day 1 P1

November 1, 2017

geometry

Problem Statement

Two circles,

\omega_1

and

\omega_2

, centered at

O_1

and

O_2

, respectively, meet at points

A

and

B

. A line through

B

meet

\omega_1

again at

C

, and

\omega_2

again at

D

. The tangents to

\omega_1

and

\omega_2

at

C

and

D

, respectively, meet at

E

, and the line

AE

meets the circle

\omega

through

A, O_1,O_2

again at

F

. Prove that the length of the segment

EF

is equal to the diameter of

\omega

.

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