Pedal Triangles

by

Stacy Musgrave

The pedal triangle is constructed as follows: given a triangle ABC and a point P, you construct the perpendicular to each side (extended if necessary) through P and connect these points of intersection. This gives you a triangle DEF, called the pedal triangle. As with most things, this is best demonstrated by picture.

Here's a picture where the pedal point P is in the interior of triangle ABC.

Here's a picture where the pedal point P is in the exterior of triangle ABC.

What happens when P is on triangle ABC?

First we look at when it's on a side, but not a vertex.

Now, we look at when P is one of the vertices A, B or C.

It's interesting to note that in this last case, the pedal triangle degenerates to a straight line.

So we ask ourselves: is this the only time that the pedal triangle is degenerate?

Further exploration suggests not. Click on the GSP file below to see when we place the pedal point P on the circumcircle of ABC, the pedal triangle is always degenerate. Challenge: Prove that DEF is always a straight line segment.

Pedal Triangle with P on the Circumcircle