Explorations in Geometry
In Exploration # 4 you constructed a triangle inside another triangle and found that the smallest perimeter occurred when the points were at the feet of the altitudes of the original triangle. This inner triangle is called a pedal triangle. Pedal, because it connects the "feet" of the altitudes. In that triangle, it may have become clear to you that the altitudes of the exterior triangle are the angle bisectors of the pedal triangle. If this was not clear, you might revisit the previous exercise, and construct the triangle connecting the feet of the three altitudes and observe that this is, indeed true. You should be able to prove that it is true, with certain limitations (what are those limitations?) This exploration extends that idea with a single altitude.