Medial Triangle Locus Theorem

Given an original triangle (thick line), find the medial triangle (outer thin line) and its incircle. Take the pedal triangle (inner thin line) of the medial triangle with the incenter as the pedal point. Now pick any point on the original triangle, and connect it to the point located a half-perimeter away (gray lines). Then the locus of the midpoints of these lines (the s in the above diagram) is the pedal triangle.