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line tangent and parallel at the same time.

Source: Iranian RMM TST 2021 Day1 P1

April 16, 2021

geometrycirclesorthocenter

Problem Statement

Suppose that two circles

\alpha, \beta

with centers

P,Q

, respectively , intersect orthogonally at

A

,

B

. Let

CD

be a diameter of

\beta

that is exterior to

\alpha

. Let

E,F

be points on

\alpha

such that

CE,DF

are tangent to

\alpha

, with

C,E

on one side of

PQ

and

D,F

on the other side of

PQ

. Let

\alpha

be the intersection of

CF,AQ

and

T

be the intersection of

DE,QB

. Prove that

ST

is parallel to

CD

and is tangent to

\alpha

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