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You have an extra bracket you don't want. This code works as you expect:

var('x x1 y y1 r r1')
solve((x-x1)^2 + (y-y1)^2 == (r+r1)^2, r)

Here's a link to a live example.

You have an extra square bracket you don't want. This code works as you expect:

var('x x1 y y1 r r1')
solve((x-x1)^2 + (y-y1)^2 == (r+r1)^2, r)

Here's a link to a live example.