Touchy Circles Part 2

Geometry Level 3

Circles \(A, B,\) and \(C\) are placed in the first quadrant so that circle \(A\) is tangent to both the \(x\)-axis and the \(y\)-axis. Circle \(B\) is tangent to the \(x\)-axis and is externally tangent to circle \(A\), while circle \(C\) is tangent to the \(y\)-axis and is externally tangent to \(A\).

If a line passes through the centers of all three circles and the radii of circles \(B\) and \(C\) are equal, then how many times larger is the radius of circle \(A\) than the radius of circle \(B\)?

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