PEDAL TRIANGLES

1a. Let triangle ABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC (extended if necessary) locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.

Above see the (red) pedal triangle to triangle ABC.

2. What if pedal point P is the centroid of triangle ABC? Well, let's see:

The pedal triangle is inside the original triangle. It is possible to have any one of the vertices of the pedal triangle outside the original triangle, but only one at a time:

The pedal triangle is only degenerate when the original triangle is degenerate. Click here to check this out. Move the vertices around and see the wonder of GSP.

3. What if pedal point P is the incenter?

The pedal triangle is inside the original triangle and cannot escape - not even a single vertex! Click here to investigate. The pedal triangle is degenerate only when the original triangle is degenerate.

4. What if pedal point P is the orthocenter?

If the orthocenter is inside the triangle, the pedal triangle is inside; if the orthocenter is outside the triangle, two vertices of the pedal triangle also are outside the triangle; if the orthocenter is at a vertex (a right triangle), then the pedal triangle is degenerate. Click here to experiment.

5. What if pedal point p is the circumcenter?

If the circumcenter C is used as the pedal point, the triangle is always inside the original triangle. This is true regardless of whether the pedal point is inside or outside the original triangle. The pedal triangle is degenerate only when the riginal triangle is degenerate.

6. What if the pedal point P is the center of the nine point circle? Here two vertices can escape the interior of the original triangle, but only two at a time.

It is also possible to get only one vertex outside the original triangle.

Interestingly, if the pedal point N is inside the original triangle, it is inside the pedal triangle; if the pedal triangle is outside the original triangle, it is also outside of the pedal triangle; if the pedal point N is on a side of the original triangle, it is at a vertex of the pedal triangle. Try it here.

7. What if pedal point P is on a side of the triangle?

Two vertices can escape, one vertex at a time. (The vertex at the pedal point cannot ever leave it's position.)

8. What if P is one of the vertices of triangle ABC?

If the pedal point is at a vertex, the pedal triangle is degenerate.

Check it out yourself here. Drag the pedal point P near and to a vertex and observe the pedal triangle's changes.

17. Construct an excircle of triangle ABC. Animate the Pedal point P about the excircle and trace the loci of the midpoints of the sides of the Pedal triangle. What curves result?

Below see the green excircle, red pedal triangle, and blue loci in question:

To investigate on your own, look here. Drag pedal point P around the green excircle. It looks like a coin flipping, or a pizza toss! Really those shapes are ellipses. Each midpoint of the side of the pedal triangle traces an ellipse. In this sketch, the ellipses appear to be the same shape but have different orientation.

Look at the angle bisectors (blue dashed lines) through the center of the excircle. How are the loci positioned with respect to the angle bisectors?

Look here. There are many lines to confuse the issue, so try pulling point P around the green circle again. It appears that the angle bisectors cut the ellipses into congruent halves.

How to prove that is another story.