The Extended Incenter of a Convex Polygon

The incenter is the unique point in any triangle where all interior angle bisectors meet. For convex polygons with more than three sides, the bisectors generally fail to be concurrent. We introduce the extended incenter, a connected structure that every interior angle bisector must cross. We prove that the region formed by a set of three anchor points is pierced by every bisector (the Piercing Theorem), that anchor triangles exist in every convex polygon, and that any connected set separating the three sides of such a triangle is a transversal structure (the Transversal Theorem). The approach adapts the classical transversal framework of Hadwiger 4 and Wenger 9 from a single line transversal to a set of structures embodying a transversal region.