through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter; every line through the incenter that splits the area in half also splits the perimeter in half. There are either one, two, or three of these lines for any given triangle. Relative distances from an angle bisector Let X be a variable point on the internal angle bisector of A. Then X = I'' (the incenter) maximizes or minimizes the ratio along that angle bisecter. References External links Triangle centers