A Closed Form for the Chord-Power Integral I_2 of a Triangle

Abstract

The chord-power integrals $I_k$ are classical integral-geometric functionals of a planar convex body, obtained by integrating powers of the chord length against the kinematic measure on the space of lines meeting the body. We establish a single-expression closed form for $I_2$ on an arbitrary triangle, involving logarithms symmetric in the sides, and derive two analytic consequences: a power-sum series representation, and a sharp isoperimetric-type inequality with explicit constant involving $\ln 3$, attained uniquely by the equilateral triangle. The set $\{I_0, I_1, I_2\}$ identifies a triangle up to congruence, complementing J. Gates's algebraic recognition via $\{I_0, I_1, I_5\}$ with the minimal index set $\{0, 1, 2\}$.