We prove three convex hull theorems on triangles and circles. Given a triangle ${ riangle}$ and a point p, let ${ riangle}^{prime}$ be the triangle each of whose vertices is the intersection of the orthogonal line from p to an extended edge of ${ riangle}$. Let ${ riangle}^{{prime}{prime}}$ be the triangle whose vertices are the centers of three circles, each passing through p and two other vertices of ${ riangle}$. The first theorem characterizes when $p{in}{ riangle}$ 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 $p{in}{ riangle}$ if and only if $p{in}{ riangle}^{prime}$. These are used to prove the third: Suppose p be does not lie on any extended edge of ${ riangle}$. Then $p{in}{ riangle}$ if and only if $p{in}{ riangle}^{{prime{prime}}$.