The Department of Mathematics and Science Education

tiffani c. knight

First, I created triangle ABC. Then I picked an arbitrary point P in the plane. The next instruction was to create a Pedal triangle. Apparently, a pedal triangle is created when one draws perpendicular lines from the pedal point, P in this case, to the three sides of the triangle. The intersections of those lines, points R, S, and T, form the Pedal triangle RST. See below:

This was pretty interesting. I played around by moving P to observe the changes in the Pedal triangle. I observed the pedal triangle when P was above and below the original triangle as well as to the left and right of it…the shapes were interesting to watch, but I didn’t pick up on any major differences.

Out of the options given, I’d like to explore what happens when P is on the circumcircle, when P is the circumcenter, when P is the orthocenter, and when P is on a side of the triangle.

So, get this: I’m exploring the circumcircle stuff that I was told to do…I constructed the circumcircle and made my point P on it and I kept getting a straight line. I just knew that I was doing something wrong, because I’m not too savvy when it comes to these things, but apparently, I was doing it right! Imagine that! : ) According to this webpage: http://en.wikipedia.org/wiki/Wik/Pedal_triangle I am supposed to get a straight line! Ha! I may be okay after all.

Here’s the diagram when P is on the circumcircle: If you will notice, LJK is not a pedal triangle, it’s a pedal line, which is also called the Simson line, according to the site. The LJK continues to be a line as you move P along the circumcircle

Here’s the diagram when P is the circumcenter:

When point p that was the circumcenter, the pedal triangle (in black) was contained within the original triangle. What’s also interesting to note is that the pedal triangle divided the original triangle ABC into 3 equal triangles. (4 if you include the pedal triangle). That’s interesting.

Here’s the diagram when P is the orthocenter:

So, when P is the orthocenter, the pedal triangle NIO was again contained within the original triangle ABC.

Here’s the diagram when P is on a side of the triangle:

When P was on a side of the triangle, the pedal triangle created contained P as a vertex. And pedal triangle MPQ was contained within the original triangle ABC.