Thomas Earl Ricks

Mathematics Education

Assignment # 9

Investigation #11b

ÒPedal Triangles and SimsonÕs LineÓ

We will review briefly what the pedal triangle is, and then explore SimsonÕs Line in more depth, especially the envelope created by the lines as the pedal point is moved about the circumcircle of the original triangle.

Pedal Point and Pedal Triangle.

If given any triangle ABC in the plane:

And any point P in the plane:

By constructing perpendiculars through P to the sides of ABC (extended if necessary)

Forms points R, S, and T respectively

Connecting these points forms triangle RST, which is a pedal triangle for the pedal point P.

Notice that we can create the pedal triangle without showing the perpendicular lines:

What happens as P moves about the plane?

In the above graphic, P is below triangle ABC

Now it is inside triangle ABC.

So the pedal triangle changes shape as P moves around.

Here is a pedal triangle for P that involves extending the sides of triangle ABC to find the points of intersection of the perpendiculars:

And here is pedal triangle for a different triangle ABC:

Here is pedal triangle that is completely outside triangle ABC:

To explore different triangles and their pedal triangles as point P moves around, click here for a GSP file to open and explore.

Question:  Can you find a pedal triangle that compresses to just a line segment?

This line segment is known as the SimsonÕs Line.  This brings us to our next topic.

## SimsonÕs Line

There are locations in the plane for P that compresses the pedal triangle into a single line segment.  When this occurs, the compressed pedal triangle is called SimsonÕs Line.

Here is an example:

We will now hide the dashed lines to find other location of P that form SimsonÕs Lines.

Three locations you may have found where this occurs is the verticies of triangle ABC:

But there are many other locations for P that yield SimsonÕs Lines.  Can you describe the places where P gives SimsonÕs Lines?  Is there a pattern to where P should be?

If you guessed or found the circumcircle, you are right!

What else can you discover about pedal points and pedal triangles?