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4 unequal circles tangent to 2 circles and a line

Source: III Soros Olympiad 1996-97 R1 11.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 29, 2024

geometrytangent circles

Problem Statement

On the plane there are two circles

a

and

b

and a line

\ell

perpendicular to the line passing through the centers of these circles. It is known that there are

4

unequal circles, each of which touches

a

b

and

\ell

. Find the radius of the smallest of these four circles if the radii of the other three are

2

3

and

6

. Also find the ratio of the radii of the circles

a

and

b