exupero's blog

Circles on a hypotenuse problem

Here's a geometry problem posed to me a while ago: given a right triangle with two congruent circles centered on the hypotenuse and tangent to each other and the legs of the triangle, and given the radii of the circles and the length of one of the triangle's legs, find the length of the triangle's other leg.

Diagrammed, the problem looks like this:


We know a and r, and we 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:


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.