Pedal Triangles

A Pedal Triangle can be constructed for a triangle, with the Pedal Point, P, in an infinite number of places on the plane of the triangle. The relationship between the location of the P and the resulting Pedal Triangle can be examined. We will focus upon the results when the Pedal Point P is located on the nine-point circle of a given triangle, as shown below.

In the preceding diagram, the green circle is the nine-point circle of the black triangle, whose sides have been extended as dashed lines. The red triangle is the Pedal Triangle of the point on the green circle. The blue circle is the circumcircle of the black triangle. The nine-point circle was chosen for the location of the Pedal Point P, because it can intersect the circumcircle of the given triangle. The nine-point circle can intersect the circumcircle in two places or it can be tangeant to the circumcircle at one point. Below are two ways that a nine-point circle can intersect the circumcircle.

Notice that if P is located upon the circumcircle, the resulting Pedal Triangle is the Simpson Line (all the vertices are colinear). If the sides of the Pedal Triangle are extended as lines, it is very easy to see that they are colinear.

These lines can be traced, as P is moved along the nine-point circle, with the following result. The narrowest point between the two sides of the curve seem to be defined when the Pedal Triangle has degenrated into the Simpson Line.

By examining the case when the nine-point circle and the Circumcircle are tangent (above), it can be seen that the width narrowest point is not bordered by Simpson Lines on both sides, because in this case there is only one Simpson Line, and the narrowest point of the path of the Simpson Lines is wider.

Geometer SketchPad document, that was used within this examination of Pedal Triangles

Geometer SketchPad script library that contains the scripts used to create the Pedal Triangle and others.

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