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So few points, so many tangencies

Source: European Mathematical Cup 2022, Senior Division, Problem 4

December 20, 2022

geometrytangentcircletrapezoid

Problem Statement

Five points

\angle ABC > 90^{\circ}

,

B

,

C

,

D

and

E

lie on a circle

\tau

clockwise in that order such that

AB \parallel CE

and

\angle ABC > 90^{\circ}

. Let

k

be a circle tangent to

AD

,

CE

and

\tau

such that

k

and

\tau

touch on the arc

\widehat{DE}

not containing

A

,

B

and

C

. Let

BF

be the intersection of

\tau

and the tangent line to

k

passing through

A

different from

AD

.
Prove that there exists a circle tangent to

BD

,

BF

,

CE

and

\tau

.

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