 Pedal Triangles

Kasey Nored

A pedal triangle is constructed by picking a point and constructing perpendicualr lines between the point and the lines along which the sides a given triangle lie. For a script tool of the Geometer's Sketchpad Construction click HERE.

Here is a Java Construction that allows you to mainipulate the Pedal Point P. The yellow triangle is known as the Pedal Triangle.

#### Pedal Triangle

To further explore the pedal triangle, lets look at a specific case, where a pedal triangle of a pedal triangle's pedal point is the incenter of triangle ABC. A few things are immediately apparent, the verticies of the second and third pedal triangles lie along the angle bisectors of the original triangle.

Let's explore some of the interesting ideas of a pedal triangle such as the Simpson Line, where the verticies of the pedal triangle are collinear. To begin this investigation we need a circumcenter. After hiding the perpendicular lines that provide us with a circumcenter, we created two Simpson lines, pedal triangles with point P on the circumcenter of the original triangle. The pedal points are labeled P1 and P2 and the intersection of the two lines upon which the pedal triangles lie is labeled S. We want to explore the relationship between the arc measure between the two pedal points and the angle measure of the intersection of the Simpson lines. As you can see here the angle measure of the s equal to twice the measure of angle BSA, which is the opposite angle from the arc. This relationship is maintained regardless of the translation of the pedal points.  