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Circles on a hypotenuse problem

Here's a geometry problem posed to me a while ago: when two congruent circles of known radius whose centers lie on the hypotenuse of a right triangle and are tangent to both each other and to their respective legs of the triangle, can the length of one leg be determined from the length of the other leg and the radii of the circles?

Diagrammed, the problem looks like this:

abrr
Two circles on the hypotenuse of a right triangle, tangent to each other and the legs of the triangle.

We know a and r, and want to find b.

At first I wasn't sure the problem was fully constrained and didn't have any degrees of freedom, but we can find an equation in terms of only a, r, and the angle at the top of the triangle—two knowns and only one unknown:

arrrrθcdd
Some additional right triangles within the diagram, which may help.
StartLayout 1st Row 1st Column a 2nd Column equals c plus 2 d plus r 2nd Row 1st Column d 2nd Column equals r cosine normal theta 3rd Row 1st Column tangent normal theta 2nd Column equals StartFraction r Over c EndFraction 4th Row 1st Column c 2nd Column equals StartFraction r Over tangent normal theta EndFraction equals r cotangent normal theta 5th Row 1st Column therefore a 2nd Column equals r cotangent normal theta plus 2 r cosine normal theta plus r EndLayout

Unfortunately, Wolfram Alpha is not optimistic about finding a simple solution for θ from this equation.

Given the five right triangles above, we can label all the sides and generate more than a dozen equalities using the sums of line segments, the ratios of similar triangles, and the Pythagorean theorem, but substituting unknowns until we get an equation with only a, r, and b has so far yielded only a complex quartic equation that doesn't seem much more solvable for b than the trigonometric equation already found.

Given the simplicity of the problem statement, it seems like there should be a simple solution, but I haven't found it yet. If you have a suggestion, please email me.