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Inscribed octagon, geometry

Source:

August 3, 2021

octagoncirclesradical axisconcurrencycommon tangentsgeometry

Problem Statement

Octagon

A_1A_2A_3A_4A_5A_6A_7A_8

is inscribed in a circle

\Omega

with center

O

. It is known that

A_1A_2\|A_5A_6

,

A_3A_4\|A_7A_8

and

A_2A_3\|A_5A_8

. The circle

\omega_{12}

passes through

A_1

,

A_2

and touches

A_1A_6

; circle

\omega_{34}

passes through

A_3

,

A_4

and touches

A_3A_8

; the circle

\omega_{56}

passes through

A_5

,

A_6

and touches

A_5A_2

; the circle

\omega_{78}

passes through

A_7

,

A_8

and touches

A_7A_4

. The common external tangent to

\omega_{12}

and

\omega_{34}

cross the line passing through

{A_1A_6}\cap{A_3A_8}

and

{A_5A_2}\cap{A_7A_4}

at the point

X

. Prove that one of the common tangents to

\omega_{56}

and

\omega_{78}

passes through

X

.

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