THREE CONVEX HULL THEOREMS ON TRIANGLES AND CIRCLES Kalantari, Bahman; Park, Jong Youll;
We prove three convex hull theorems on triangles and circles. Given a triangle and a point p, let be the triangle each of whose vertices is the intersection of the orthogonal line from p to an extended edge of . Let be the triangle whose vertices are the centers of three circles, each passing through p and two other vertices of . The first theorem characterizes when via a distance duality. The triangle algorithm in  utilizes a general version of this theorem to solve the convex hull membership problem in any dimension. The second theorem proves if and only if . These are used to prove the third: Suppose p be does not lie on any extended edge of . Then if and only if .
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