An Exploration of Pedal Triangles

By

Ken Montgomery

First, we constructand then we construct lines through each side of. We then create a pedal point P, through which we construct lines that are perpendicular to the sides of. The intersections of these lines are the vertices of the pedal triangle, (Figure 1). For a tool to construct a pedal triangle, download the toolbox in Assignment 5.

Figure 1: Pedal triangle, forand pedal point, P

In our first investigation, we let the pedal point, P be the Centroid of(Figure 2).

Figure 2: P is the Centroid of

We find that the vertices ofeach lie on the sides of. In our second exploration, we let the pedal point, P be the Incenter of (Figure 3).

Figure 3: P is the Incenter of

By constructing the triangle centers for , we see that when P is the Incenter of, P is also the Circumcenter of. Thirdly, we let P be the Orthocenter of(Figure 4).

Figure 4: P is the Orthocenter of

A similar exploration of the triangle centers ofyields the realization that A is now the Incenter of. In our fourth investigation, we let P be the Circumcenter of(Figure 5).

Figure 5: P is the Circumcenter of

Again, we have the vertices of, each on the sides of. We see that the Orthocenter ofis now, also the Circumcenter of. We see, also that the Centroid ofis now also the Centroid of. To explore these properties of the pedal point and the pedal triangle, download Assign92KM.gsp. Lastly, we construct the Circumcenter ofand a circle with radius larger than that of the Circumcircle of (Figure 6).

Figure 6: Construction of circle with radius greater than that of the Circumcircle of , but centered at the Circumcenter of

We then let the pedal point, P be a point on the circle and animate the point by rotation around the circle. In this last investigation, we turn our attention to the loci of the midpoints of the pedal triangle, as P follows this circular path. In doing so, we find that all three loci are ellipses (Figure 7).

Figure 7: The loci of the midpoints of the pedal triangle, as P rotates in a circular path, are elliptical

Download Assign9KM.gsp to further explore the resulting loci of these midpoints. By changing the radius of the circle, for instance the resulting loci retain their elliptical shapes, but are dilated proportionally (Figure 8).

Figure 8: The circular path of P is the Circumcircle for

What is also interesting in this dilation is that when the circular path is the Circumcircle of, each of the elliptical loci is tangent to the Circumcircle and each point of tangency is a vertex of. To experiment with this configuration, download Assign911KM.gsp.