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. We prove three convex hull theorems on triangles andcircles. Given a triangle △and a point p, let △ ′ be the triangle eachof whose vertices is the intersection of the orthogonal line from ptoan extended edge of △. Let △ ′′ be the triangle whose vertices arethe centers of three circles, each passing through p and two othervertices of △. The rst theorem characterizes when p ∈ △ viaa distance duality. The triangle algorithm in [1] utilizes a generalversion of thistheorem to solve the convex hullmembership problemin any dimension. The second theorem proves p∈△if and only ifp∈△ ′ . These are used to prove the third: Suppose pbe does notlie on any extended edge of △. Then p∈△if and only if p∈△ ′′ . 1. IntroductionWe begin with three gures in order to describe three convex hullproperties of a general triangle to be proved here.In Figure 1 we have a point p inside a triangle △ABC. We havedrawn three circles each centered at a vertex passing through p. Notethat the three corresponding discs intersect only at p. Will this propertyalways be valid? What happens if pis outside of △ABC?We mention that a well-known theorem, “Johnson’s three circle the-orem” is a result on three circles of equal radius r passing through acommon point: The circle that passes through the three centers also hasradius r. Johnson’s Theorem is visually evident. However, a naturalquestion is to characterize the cases when the circles do pass through acommon point. This characterization will follow here as a special case.

THREE CONVEX HULL THEOREMS ON TRIANGLES AND CIRCLES