Assignment 9: Pedal Triangles

by

Mike Cotton

Given a triangle and an arbitrary point P, a pedal triangle can be constructed.


Figure 1

The construction of the pedal triangle RST begins with constructing perpendiculars to the sides of ABC from point P. The points there the perpendicular lines intersect the sides of triangle ABC are the points R,S, and T. Where R, S and T are the vertices of the pedal triangle. Click on the image above to manipulate point P and the size/shape of the triangle.
Now you ask what is the significance of the pedal triangle with different positions of P with respect to the triangle ABC. Let's look at the case when point P coincides with the orthocenter of triangle ABC.


Figure 2

The figure above (Figure 2) shows triangle ABC with its orthic triangle within (blue). Click on the image to move point P so it coincides with point H (orthocenter of triangle ABC). The pedal triangle will coincide with the orthic triangle when the pedal point P coincides with point H. This is true even if triangle ABC is obtuse.
Let's look at the case when point P coincides with the circumcenter of triangle ABC.


Figure 3

In the figure above (Figure 3) the point C is the circumcenter of triangle ABC. Click on the image to move pedal point P over point C. What does the pedal triangle seem to represent? If you said the medial triangle you are correct. The figure below is an image of triangle ABC with its medial triangle. Click on the image to confirm whether the pedal triangle forms the medial triangle when the pedal point P coincides with the circumcenter (C).


Figure 4

In fact, the pedal triangle does indeed form the medial when the pedal point P coincides with the circumcenter of triangle ABC. Even if triangle ABC is obtuse.

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