Brenda King

Pedal Triangles

Introduction:

Pedal triangles are constructed by starting with any triangle, ABC and a 4th point located anywhere in the plane.  The vertices to pedal triangle, RST, are formed from the intersection of perpendiculars to the sides of triangle ABC (extend sides if necessary) where the perpendiculars also pass through the point P. See diagram 1.

Diagram 1.

This investigation will explore what happens when the point P is relocated to coincide with the Circumcenter of triangle ABC.

Point P used as circumcircle for ∆ABC

The circumcenter of a triangle is the point in the plane equidistant from the three vertices of a triangle.  Since all points equidistant from two points lie on the perpendicular bisector of the segment determined by the two points, the circumcenter is located on the perpendiculars bisector of each side of the triangle.

From the way the Pedal triangle was constructed, with perpendiculars at the vertices of the triangles, moving the point P to the circumcenter also moves the perpendiculars to the midpoints.  The Pedal triangle, RST, has turned into a medial triangle!

Diagram 2

The medial triangle has unique characteristics of its own.  The medial triangle can be viewed as similar to the image of ABC.  To explore this idea, I rotated RST 180° around the centroid of RST.  The Centroid of a triangle is the common intersection of the three medians.  Next I dilated RST by a factor of 2.  See diagram 3

Diagram 3

It also follows from this that the perimeter of the medial triangle equals the semiperimeter of triangle ABC, and that the area is one quarter of the area of triangle ABC.  I checked out these properties in Geometry Sketchpad as shown in Diagram 4

Diagram 4