# Random chords of two circles and a third centre

Open problem settled, formally verified in Lean 4. Vamshi Jandhyala.

> Lines AB and BC hit the third circle equally often, for every ratio of radii in geometric progression.

Canonical: https://vamshij.com/research/random-chords-two-circles
Code and Lean proofs: https://github.com/jvvk/mathematics/tree/main/random-chords-two-circles
Paper (PDF): https://github.com/jvvk/mathematics/blob/main/random-chords-two-circles/paper/fig_chords.pdf

Three circles touch in a row with radii `a`, `b`, `c` in geometric progression. Choose `A` uniformly on the first circle and `B`, `C` uniformly on the second. Dan observed numerically, and asked on MathOverflow (question 499477) why, that the lines `AB` and `BC` meet the third circle with the same probability.

The paper explains it through a fact about two circles. For independent uniform points `A`, `B` on two circles whose discs have disjoint interiors, the signed offsets of the line `AB` from the two centres, each divided by its radius, are independent with the arcsine law (Lemma 2). The proof is a change of variables over the four pairs of points on a line, whose contributions add to a constant because the half-chords cancel. From this, Theorem 4 decides exactly which points `O` on the line of centres see `AB` and the chord `BC` at the same distance in distribution: `O` is the second centre, or the circles touch and `|QO| = b(a + b)/a`. This answers Dan's question for every ratio, extends it to every circle about the third centre and to chains of circles, shows that the progression is necessary, and gives a single integral for the common hit probability.

Preprint v1, 8 October 2026, not peer reviewed. Every proved result of the paper is also formally verified in Lean 4 (`lean/`, standard axioms only). The exposition has not yet been independently reviewed. The author used AI tools (Claude, Anthropic; Codex, OpenAI) in this work, as described in the paper's acknowledgements, and is responsible for its content.