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?

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

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

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

What if pedal point P is the incenter of 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.