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Three pairwise externally tangent circles

\omega_1,\omega_2

and

\omega_3

are given. Let

K_{12}

be the point of tangency between

\omega_1

and

\omega_2

and define

K_{23}

and

K_{31}

similarly. Consider the point

A_1

\omega_1

. Let

A_2

be the second intersection of the line

A_1K_{12}

with

\omega_2

. The line

A_2K_{23}

then intersects

\omega_3

the second time at

A_3

, and then line

A_3K_{31}

intersects

\omega_1

again at

A_4

and so on.
[*]Prove that after six steps, the process will loop; that is,

A_7=A_1

. [*]Prove that the lines

A_1A_2

and

A_4A_5

are perpendicular. [*]Prove that the triples of lines

A_1A_2,A_3A_4

and

A_5A_6

and

A_2A_3,A_4A_5

and

A_6A_1

intersect at two diametrically opposite points on the circle

(K_{12}K_{23}K_{31})

.
Proposed by E. Morozov

Problem Statement