Philippa M. Rhodes

Write-up 9

Select any triangle, ABC. If P is any point in the plane ,
then the three points of intersection, R, S, and T, formed by
constructing perpendicular lines to the sides of ABC

locate the vertices of the Pedal Triangle. Triangle RST is the
Pedal Triangle for Pedal Point P.

Click **here** (for the GSP file)
to move the Pedal Point P or to change Triangle ABC.

What if pedal point P is the centroid of triangle ABC?

Click **here** to change Triangle
ABC.

What if pedal point P is the incenter of triangle ABC?

Click **here** to change Triangle
ABC.

What if pedal point P is the orthocenter of
triangle ABC?

Click **here** to change
Triangle ABC.

What if pedal point P is the circumcenter of triangle ABC?

Click **here** to change
Triangle ABC.

What if pedal point P is the incenter of triangle ABC?

Click **here** to change
Triangle ABC.

**GOAL:** **Find all conditions in which the three vertices
of the Pedal Triangle are colinear. This line is called the Simson
Line.**

We see that the three vertices of the pedal triangle are collinear
when the pedal point is one of the vertices of Triangle ABC.

By moving the pedal point slowly to various locations so that
the three vertices of the pedal triangle remain collinear, the
path appears to be a circle. Well, it is the circumcircle of triangle
ABC.

Thus, anytime the pedal point is on the circumcircle
of triangle ABC, then the three vertices of the pedal triangle
are collinear.
Click **here** for a GSP
animation of the pedal point as is moves around the circumcircle.

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