The Pedal Point in Various Positions and Discussion of the Resuting Pedal Triangles

by: Al Byrnes

What is a pedal point? What is a pedal triangle? How can we relate this new concept to some of the other concepts that we have already discussed, specifically, the concept of medial triangles, reviewed in exploration number 6? Continue to find out more:

The pedal triangle is "the triangle whose polygon verticies are the feet of the perpendiculars from P to the side lines" (Weissman, n.d.). Let's gain some more insight into this formal definition with a walkthrough of the pedal triangle's construction in Geometer's Sketchpad:

First, let's begin with the triangle ABC:

Next, let's construct a point P. This point will be the pedal point. The property the pedal point must satisfy from Weissman's definition above is the segments connecting the pedal point to the legs of the triangle must be perpendicular to those triangle legs:

We can be sure the segments connecting P to the three legs of the triangle ABC are perpendicular to the legs of the triangle by the nature of our construction (the three segments were all constructed by selecting the pedal point and a triangle leg and using the "construct - perpendicular line" command in GSP).

Finally, we will construct the pedal by connecting the three verticies created by the intersections of the triangle legs and the perpendiculars, originating at P. These three verticies are referred to as the points S, R, and T in the illustration below:

And there you have it, a pedal triangle.

But what if the pedal point lies outside the triangle itself? After all, the line segments that represent the legs of the triangle AB, BC, and AC all lie on lines. When we represent this triangle in this way, we will see that a pedal triangle can be consrtucted for any point in the plane of ABC. Here are some examples to illustrate this point, along with a GSP script tool so you can investigate for yourself.

Examples of pedal triangles, with P in a variety of locations in the plane of triangle ABC:

P inside of the triangle:

P outside of the triangle ABC, between lines that contain the segments AB and AC:

P outside of the triangle ABC, between the lines that contain the segments AB and BC:

P outside of the triangle ABC, between the lines that contain the segments AC and BC:

P outside of the triangle ABC, between the lines that contain the segments AC and AB:

P outside of the triangle ABC, between the lines that contain the segments BC and AB:

P outside of the triangle ABC, between the lines that contain the segments BC and AC:

Find the Geometer's Sketchpad file here for more exploration.

So, pedal triangles for the triangle ABC will exist both inside and outside of the triangle ABC. What about some other potentially interesting cases discussed in other explorations? In write-up 4, we investigated the centriod, e.g.: one of the many centers of the a triangle. The centriod was formed by connecting each of the midpoints of the legs of the triangle with the vertex opposite of each triangle leg. For a refresher, see the illustration of the centroid G for the triangle ABC below:

Now let's see what happens if the pedal point P corresponds with the centroid G. Do you have a guess? I think that the pedal triangle will not correspond with the triangle of medians formed by connecting the midpoints of the triangle ABC! Triangle of medians:

Triangle formed by placing the pedal point P at the centroid:

Note: when P and G were merged, the medial triangle did not form the medial triangle as I had predicted that it wouldn't. Why do you think this happened? Hint: are the segments that are used to form te centroid perpendicular to the legs of the triangle? How about the segments which connect the pedal point P to the legs of the triangle ABC?

There is a type of center for a triangle, the orthocenter, which might have a better chance of forming a triangle, which would match the pedal triangle. This triangle is called the orthic triangle.

The first illustration below is the triangle ABC shown with the orthocenter H and the shaded region, representing the orthic triangle (constructed using the verticies formed by the points on the legs of the triangle that are coincident with the perpendicular bisectors that originate from the vertex at the opposite side of the triangle). Below: orthocenter and orthic triangle:

Now merge the orthocenter point H with the pedal point P and see what the resulting pedal triangle looks like. I think that the resulting pedal triangle will be the same orthic triangle we see above. Thoughts? Why might that be? Below: triangle ABC with pedal point P at the same point as H. To emphasize the point, the orthic triangle constructed above will be left shaded and the pedal triangle will be drawn with no shading and outlined in red:

Sure enough, it appears that our intuition did not fail us. However, I will emphasize that this GSP sketch is not intended to be a substitute for a proof, but rather a suggestion that a proof showing that the orthic triangle of ABC and the pedal triangle of ABC with P at H are congruent would be an appropriately justified next step, as this seems to be the case.

Finally, I wanted to examine some other exceptional locations for the pedal point P and make some conclusions about the resulting pedal triangles. Firstly, let's consider the case when P lies on a side of the triangle:

Click here and animate the point P (which has been merged with R) to see what happens to the pedal triangle as it moves along the leg of the triangle ABC, defined by BC. This illustration becomes even more interesting once the pedal point becomes coincident with one of the verticies B or C. See below:

Now, the pedal triangle has become and altitude of the triangle ABC.


Works cited:

Weisstein, E. W. "Pedal Triangle." From MathWorld - A Wolfram Web Resource.

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