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3 tangents circles in pairs, similar triangles, angles wanted

Source: Austrian-Polish 1986

May 24, 2019

geometrysimilar trianglestangent circlesangles

Problem Statement

A non-right triangle

A_1A_2A_3

is given. Circles

C_1

and

C_2

are tangent at

A_3, C_2

and

C_3

are tangent at

A_1

, and

C_3

and

C_1

are tangent at

A_2

. Points

O_1,O_2,O_3

are the centers of

C_1, C_2, C_3

, respectively. Supposing that the triangles

A_1A_2A_3

and

O_1O_2O_3

are similar, determine their angles.