$ABC $ is an isosceles right triangle, whose small sides are $1$. I want to find a close form for the sum of the infinite radii of the small circumferences created by the cevians. The numerical value of this sum is $ 0.39709... $
Furthermore what kind of curve pass by the centers of the circles?
Using the formula for the inradius of a triangle it is easy to compute the $n$-th radius: $$ r_n={1\over 1+n\sqrt{n^2+2n+2}+(n+1)\sqrt{n^2+1}}. $$ The sum $\sum_{n=1}^\infty r_n$ is finite but Mathematica doesn't give a closed form for it, hence I doubt a nice formula can be found.