Uniform Triangles with Equality Constraints

TL;DR

This paper explores the properties of uniform triangles under various equality constraints, revealing connections to Gaussian triangles and providing insights into their angle distributions and moments.

Contribution

It introduces new geometric constraints on triangles and uncovers their probabilistic relationships with Gaussian triangle angles.

Findings

The bivariate density for angles matches that of 3D Gaussian triangles.

The angle gamma opposite side c is uniformly distributed between 0 and pi.

Several side moments are derived, with some closed-form expressions still open.

Abstract

The equality constraint a+b+c=1 for random triangle sides corresponds to breaking a stick in two places. An analog a^2+b^2+c^2=1 has a remarkable feature: the bivariate density for angles coincides with that for 3D Gaussian triangles. Interesting complications also arise for a+b=1 and for a^2+b^2=1, with the understanding that the angle gamma opposite side c is Uniform[0,pi]. Closed-form expressions for several side moments remain open.