The perimeter of a triangle is the sum of its sides. This Demonstration draws a gray line that halves the perimeter of the triangle. If and only if that line halves the area, then it goes through the incenter point in red. If the line goes through a green midpoint, then it is called a cleaver and goes through the Spieker center in green. If the line goes through a vertex, then it is called a splitter and goes through the Nagel point in blue. As you move the gray line around, the midpoint in magenta traces the outline of a a smaller triangle.