## EMAT 6680 Assignment #9

## Pedal Triangles

#### By Kent Wiginton

In this assignment we will explore pedal triangles. First we
want to construct a pedal triangle. Since we have a great library
of GSP scripts we can just use them. However not everyone has
a wonderful GSP library so we will revisit how to construct a
pedal triangle. We will be constructing a pedal triangle from
triangle ABC and any point p in the plane.

Next we will construct the perpendiculars from point p to each
of the sides of the triangle. Note that depending on the triangle
we may need to extend the sides so that the perpendiculars will
intersect the lines.

The last thing we will do is make the pedal triangle itself.
The points that we are interested in are the three intersections
of the perpendiculars and the lines that go through the edges
of triangle ABC. The Pedal triangle will be colored in green.

Thus now we have a pedal triangle EFG.

Click **here** to play with a
working example in GSP, or click **here**
for a script for the general construction of a pedal triangle
to triangle ABC where P is any point in the plane of ABC.

What happens when if pedal point p is the centroid of triangle
ABC?

Recall what the orthic triangle is. The orthic-triangle is
made by connecting the feet of the perpendiculars.

Well if we were to consider H as our P in the pedal triangle
then our pedal triangle would be the exact one that we see here.
Notice that if we were to constuct the perpendiculars to each
of the sides then we will get E, F, and G. Since this is the way
that we constructed them in the first place anyway. We can see
this very easily when we constuct both on the same triangle.

Notice as P approaches the orthocenter the triangles begin
to line up. Also notice that the perpendiculars are beginning
to line up as well.

**Click here for an animation**
to see more clearly.

**Return**