Explorations in Geometry

with the

Geometerís Sketchpad

Exploration 5


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.

    1. Construct a triangle ABC and an altitude to vertex A with foot at D. Pick a point, P, on the altitude, and draw rays BP and CP cutting the opposite sides at points G and F respectively. Observe the relationship between angles FDA and GDA.
    2. Can you figure out why this is true? Can you prove this is true?
    3. Explore this figure and see if you can find any other interesting relationships (there are some interesting circles to explore here???)
    4. What other patterns and relations can you discover about this shape?
    5. Write a brief paragraph and describe the relationships you have observed in this exploration.