# Three tangent circles and a fair coin

Open problem settled, verified by computation. Vamshi Jandhyala.

> A triangle through random points on three touching circles contains the incentre with probability exactly 1/2, for all radii.

Canonical: https://vamshij.com/research/tangent-circles-incentre
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/tangent-circles-incentre
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/tangent-circles-incentre/paper/note.pdf

Three circles touch in pairs, and a triangle is formed by choosing one uniform random point on each. Dan asked on Mathematics Stack Exchange and MathOverflow ([question 498968](https://mathoverflow.net/q/498968)) why the triangle contains the incentre of the triangle of centres with probability `1/2`; the equal-radius case had been done by an integral, the unequal case only by simulation.

The probability is `1/2` for all radii (Theorem 1). The key is a fact about two touching circles: a random chord joining them crosses the common tangent at a point from which the two centres are seen at an angle uniformly distributed on `(π/2, π)` (Theorem 2). Each side of the triangle then misses the incentre with probability equal to the opposite angle of the triangle of centres divided by `2π`, and these angles sum to `π`. A refinement maps the pair of random points to a uniform point on a sphere, on which each miss is a lune (Theorem 3). The proof of Theorem 2 is a computation; the intuitive proof that the question asked for remains open.

Preprint v1, 9 October 2026, not peer reviewed. The algebra and calculus (chord identities, the integral equal to `π`, the crossing law and its distribution function, the half-angle identity, the final sum, the sphere's folding step) are formally verified in Lean 4 (`lean/`, standard axioms only); the change of variables as a statement about measures and the plane geometry of Lemmas 1 and 2 are checked by hand and by simulation. The author used AI tools (Claude, Anthropic) in this work, as described in the paper's acknowledgement, and is responsible for its content.