THREE CONVEX HULL THEOREMS ON TRIANGLES AND CIRCLES

Abstract

We prove three convex hull theorems on triangles and circles. Given a triangle <TEX>${\triangle}$</TEX> and a point p, let <TEX>${\triangle}^{\prime}$</TEX> be the triangle each of whose vertices is the intersection of the orthogonal line from p to an extended edge of <TEX>${\triangle}$</TEX>. Let <TEX>${\triangle}^{{\prime}{\prime}}$</TEX> be the triangle whose vertices are the centers of three circles, each passing through p and two other vertices of <TEX>${\triangle}$</TEX>. The first theorem characterizes when <TEX>$p{\in}{\triangle}$</TEX> via a distance duality. The triangle algorithm in [1] utilizes a general version of this theorem to solve the convex hull membership problem in any dimension. The second theorem proves <TEX>$p{\in}{\triangle}$</TEX> if and only if <TEX>$p{\in}{\triangle}^{\prime}$</TEX>. These are used to prove the third: Suppose p be does not lie on any extended edge of <TEX>${\triangle}$</TEX>. Then <TEX>$p{\in}{\triangle}$</TEX> if and only if <TEX>$p{\in}{\triangle}^{{\prime{\prime}}$</TEX>.