BY: Sharren M. Thomas

ASSIGNMENT 9

PEDAL TRIANGLES

I begin by
constructing the pedal triangle RST below.

When the pedal
point is the same as the Centroid or the incenter then the pedal triangle is
located in the interior of the original triangle.

When P corresponds
to the orthocenter of a triangle various scenarios occur. Below the pedal
triangle dissipates into a line with one of the vertices R at the intersection
of the orthocenter and the original triangle. See below

When P is the
orthocenter of an equilateral triangle then the pedal triangle is the medial
triangle of the larger triangle. See below

When P is the
circumcenter of a triangle the following observations follow. When P is
the circumcenter of an equilateral triangle, the pedal triangle will be the
medial triangle for the larger triangle. See below.

In addition, if
the pedal point P is the circumcenter of a triangle that is obtuse or acute,
the pedal triangle will still be the medial triangle, see both below.
Additionally, it appears that when P is inside the triangle the pedal triangle
is acute, when it is on the side of the triangle the pedal triangle is right
and when P is outside the triangle the pedal triangle is obtuse. In all
cases, the pedal triangle remains inside the original triangle.