Explorations
of Pedal Triangles

By Lauren Lee

What is a pedal triangle? Let triangle ABC be any triangle. If
P is any point in the plane, then the triangle formed by constructing
perpendiculars to the sides of ABC is the pedal triangle for the point P.

Let’s look at a construction:

I have labeled the points of
intersection R,S, T. So triangle RST is
the pedal triangle for the pedal point P

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What
if P is the orthocenter of triangle ABC?

We notice that P lies on the
angle bisectors of the pedal triangle RST

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What
if the orthocenter is outside of triangle ABC?

The pedal triangle RST still
lies inside triangle ABC

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What
if P is on the side of triangle ABC?

We can see that P becomes one of the vertices, in this case T, of the pedal triangle RST

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

We will notice that the
pedal triangle RST collapses. We are
left with one of the perpendiculars of pedal point P.

Let’s do one final investigation. What if P was the incenter of triangle ABC?

We notice that P is the
circumcenter for the pedal triangle RST