Assignment 9: Playing with Pedals

By Krista Floer

Here we will investigate how changing the placement of the Pedal point P changes the Pedal Triangle. GSP files are provided after each example.

CENTROID

If the Pedal point P is the centroid of the triangle, the pedal triangle will be inside triangle ABC as long as all angles are acute. I found that the pedal triangle may lie inside the triangle if an angle is obtuse, but triangle ABC can be moved so that the pedal triangle still lies inside triangle ABC.

I found that the pedal triangle may lie inside the triangle if an angle is obtuse, but the sides of triangle ABC can be changed so that the pedal triangle still lies inside triangle ABC.

It makes sense the the pedal triangle may not always lie inside triangle ABC because the perpendicular line that goes through P may hit the line containing the side of a triangle outside of the actual triangle. As seen in the picture above, the perpendicular line to AB that goes through P is outside of the triangle.

Click HERE for the GSP file of the Centroid.

INCENTER

If the Pedal point P is the incenter of the triangle, then the pedal triangle is always inside triangle ABC. This is because the incenter is always inside the triangle.

Consider the incircle. The incircle is constructed by the perpendiculars to each side going through the incenter. This is the same way the pedal constructed. So it follows that the pedal triangle would lie one the incircle. When P lies on the incenter of triangle ABC, the incircle of ABC is the circumcircle for the pedal triangle.

Click HERE for the GSP sketch of the Incenter.

ORTHOCENTER

If the Pedal point P is the Orthocenter of the triangle, then as long as the orthocenter is inside the triangle ABC the Pedal triangle will be inside triangle ABC. As soon as the orthocenter exits the triangle ABC, the Pedal Triangle is also no longer contained inside triangle ABC.

Orthocenter inside the triangle | Orthocenter outside the triangle |
---|---|

The reasoning for why this is lies in how the orthocenter is constructed. Because the orthocenter is found by the perpendiculars to a side of a triangle through a point (specifically the vertex) and the Pedal triangle is found by the perpendiculars to a side of a triangle through a point (specifically P), it follows that the orthocenter and the pedal triangle would have similar properties.

Click HERE for a GSP file of the Orthocenter.

CIRCUMCENTER

If the Pedal point P is the circumcenter of the triangle, then the Pedal triangle is always inside triangle ABC. It is also the medial triangle of triangle ABC. This holds true whether P is inside or outside triangle ABC. Here are some illustrations of this.

Circumcenter inside the triangle | Circumcenter outside the triangle |
---|---|

To understand why this happens, let's consider how the circumcenter is constructed. The circumcenter is constructed by the perpendicular bisectors of each side of the triangle. Since the Pedal point is also formed by perpendicular lines, the constructions will not change. Since each vertex on the Pedal triangle in a midpoint of a side of the triangle ABC, then the pedal triangle will also always be a medial triangle of triangle ABC.

Click HERE for a GSP file of the Circumcenter.