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Bill Tankersley's EMT 668 Page

Write-up 9

Pedal Triangles

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Let triangle ABC be any triangle. Then is P is any point in the plane, the triangle formed by constructing perpendiculars to the sides of ABC (extended if necessary) locate three points R,S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.

Click here to download a GSP script for the general construction of a pedal triangle RST to triangle ABC where P is any point in the plane of ABC.

Click here to download a GSP sketch of a pedal triangle RST to triangle ABC where P is any point in the plane of ABC.

Below is a drawing of a Pedal Triangle RST to triangle ABC for Pedal Point P:

For this write-up, I would like to concentrate on the following question:

What if P is one of the vertices of triangle ABC?

To aid in this investigation, I took the above drawing and moved Pedal Point P so that it coincides with vertex A:

After moving Pedal Point P to coincide with vertex A, I measured angle BSA, which turned out to be 90 degrees. Therefore, I think that if Pedal Point P coincides with a vertex of triangle ABC, the Pedal triangle collapses into the altitude from the vertex to the opposite side. Let's see if we get the same results when moving Pedal Point P to coincide with vertex C:

Again, we see the collapsing of the pedal triangle and that measure of angle BRT is 90 degrees. The other case would be to move Pedal Point P to coincide with vertex B, which will be left to the reader. After downloading the GSP file for the pictures above, click and drag Pedal Point P to coincide with vertex B and measure the angle formed.

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