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