Assignment 9 continued: 

Pedal Triangles & the Simson Line

by Kristina Dunbar, UGA

This page contains pedal triangles for special cases when the pedal point is on a side of the original triangle or at a vertex.  We also discuss Simson Lines.

What happens when the pedal point P is on a side of triangle ABC?

Here's what it looks like:

The above case is for an acute triangle.  When the triangle is obtuse, it looks like this:

When the original triangle is right, the pedal triangle looks like the one below:

The pedal triangle is also a right angle!

Click here for a GSP construction of the above cases.

What happens when the pedal point P is on a vertex of triangle ABC?

Here's what it looks like:

When the pedal point P is on a vertex of the original triangle, all of the vertices of the pedal triangle are collinear. 

A Simson Line is when all three vertices of the pedal triangle are collinear, forming a degenerate triangle.

When the pedal point is on a vertex of the original triangle, it does not matter what shape the original triangle is, the pedal triangle will always form the Simson Line.  In the case where P is on a vertex, the Simson Line will always be an altitude of the original triangle.  However, there are other cases in which the pedal triangle is a Simson Line, and it does not always have to be an altitude.

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