The Pedal Point of a Triangle

(is Not a Pedal at All - or is it?)

When I think of a pedal, it is usually as it relates to a bicycle. You can see in this picture that the man is making his bicycle move by pedaling. Would the bike move if there was no pedal? Is the pedal necessary? What would happen if the pedal did not move at all? My questions lead me to wonder how the pedal point of a triangle moves it. Does the pedal point move the triangle?

The pedal point of a triangle is a random point that may lie inside, outside, or on the triangle. The pedal triangle is then created by constructing 3 lines through the pedal point perpendicular to each side. The intersections of the perpendicular lines and the sides of the triangle at the right angle are then connected to form the pedal triangle.

We are going to look at various positions of the pedal point in relation to the original triangle. What happens to our pedal triangle if the point is inside the original triangle? outside? on? Let's get busy and find out!

It appears as though the closer the pedal point gets to a side of the original triangle, the larger the angle in the pedal triangle that P is moving towards. This allows the pedal triangle to be acute, right, or obtuse within the original triangle if the pedal point is inside the triangle. My original triangle was acute. I wonder if this has an effect on my pedal triangle. Will it be the same for an obtuse or right triangle?

Click here to play with the pedal point, P, of an obtuse triangle.

Click here to play with pedal point, P, of a right triangle.

What would happen to the pedal triangle if the pedal point lay on a vertex of the original triangle? Would we have an acute triangle, obtuse triangle, or right triangle? Would we have any triangle at all? It seems like the pedal point would cause the triangle to collapse if it lies on a vertex because there cannot be a perpendicular line through the point and the side at those two sides. Let's see:

How about that?! The pedal triangle collapses into a line. More importantly than just a line, it collapses into the altitude from the pedal point vertex to the opposite side. Very interesting.

What would happen if we started with an equilateral triangle? Does there exist an equilateral pedal triangle?

Of course it does! The pedal point of the equilateral triangle is the intersection of the altitudes. This causes the pedal triangle to be equilateral. I thought it might be nice to be as creative as the guy who created this picture:

This pattern of triangles was generated by taking the specific pedal point of each equilateral triangle that lay on the intersection of the altitudes. While it may not be as creative as the arranged fruit in the still life art above, it is my best attempt at creativity.


Back to the investigation. Now, let's look at the pedal triangle in the event that the pedal point is on the outside of the original triangle.

So far, I am not finding anything that is very interesting. It is interesting that when the pedal point lies on the original triangle, it is a vertex of the pedal triangle as shown:

The pedal triangles constructed here are all obtuse. Will it always be obtuse?

So, you can see that the pedal point does not move the original triangle, but the pedal triangle is highly dependent on its position - much like the bicycle's position is dependent on the pedal of the bike.


If you are still interested in the pedal point and the pedal triangle and their relationship to the original triangle, here are some other investigations in which you may look. What if the pedal point is actually the centroid of the original triangle? incenter? orthocenter? circumcenter?