Assignment 9

by

Johnie Forsythe

Pedal Triangles

Let triangle ABC be any triangle. Then if P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC (extended if necessary) locate the three points R,S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.

Click HERE to use a GSP Script tool I created to explore Pedal Triangles

Let's explore pedal triangles if the pedal point P is a center of a triangle.

What if Pedal Point is coincident with the centroid of triangle ABC?

The centroid of a triangle is the common intersection of the three medians. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.

It appears that if the pedal point is coincident with the centroid of a triangle, the pedal triangle wil similar to the larger triangle, dividing the triangle into four separate similar triangles.

What if the Pedal Point is coincident with the orthocenter of triangle ABC?

The orthocenter of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side. (Note: the foot of the perpendicular may be on the extension of the side of the triangle.)

What if the Pedal Point is coincident with the circumcenter of triangle ABC?

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

What if the Pedal Point is coincident with the incenter of triangle ABC?

The incenter of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then I must be on the angle bisector of each angle of the triangle.