ASSIGNMENT # 9


Problem # 2

 

Suppose that point P is some point in the Euclidean Plane and P is exterior to triangle ABC. A pedal triangle may be formed by constructing lines through point P which are perpendicular to each of the sides of triangle ABC, finding the points of intersection, and constructing a new triangle through these points.

It is sometimes necessary to extend the sides of the triangle in order to construct the lines perpendicular to the sides of triangle ABC. In the diagram above, the points of intersection are labeled F, G, and H. It is rather simple to use Geometer's Sketchpad to construct a triangle through these points of intersection (shown in blue).


Since P is an arbitrary point, it is necessary to try different locations for P and then see how the location of P affects the size or shape of the pedal triangle. Let's place P closer to triangle ABC and construct a pedal triangle.

 

Notice that because P was closer to triangle ABC (in pink) this time, the pedal triangle has become smaller.


In the two investigations above, point P was located outside the triangle and at different distances from one of the sides of triangle ABC. At this point, let's allow P to be a point located on one of the sides of triangle ABC and take note of how the pedal triangle is constructed.

In this case, all three vertices of pedal triangle PQR were located on various sides of triangle ABC. Moreover, pedal triangle PQR was contained in the interior of triangle ABC.


Suppose the point P is located in the interior of triangle ABC. How will this affect the attributes of the pedal triangle?

Again, the vertices of triangle RST are located on the sides of triangle ABC and the pedal triangle is located in the interior of triangle ABC.


What happens if P is the centroid of triangle ABC?

As before, the vertices of triangle RST are located on the sides of triangle ABC.


CONCLUSIONS


If point P is located outside triangle ABC, then the pedal triangle is located outside triangle ABC.

If point P is located on one of the sides of triangle ABC, then the vertices of the pedal triangle are located on the sides of

triangle ABC and the pedal triangle is located in the interior of triangle ABC.

If point P is located in the interior of triangle ABC, then the vertices of the pedal triangle are located on the sides of triangle

ABC and the pedal triangle itself is located in the interior of triangle ABC.


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