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.

Back to
Johnie Forsythe's EMAT 6680 Webpage