Pedal
Triangles

by Kasey Nored

A petal
triangle is created by the picking of a point and constructing perpendicular
lines between the point and the sides of a triangle which can be extended as
necessary.

Our beginning triangle is ABC which is green and the pedal triangle is in
yellow. Point P is our pedal point.

To further
explore the pedal triangle we are looking at a pedal triangle of a pedal
triangle of a pedal triangle where point P is the incenter of triangle ABC.

A few
things jump out immediately such as the vertices of the second and third pedal
triangle lie upon the angle bisectors of triangle ABC.

To explore
the Simpson Line, the line where the vertices of the pedal triangle are
collinear we need a circumcenter.

After
hiding the perpendicular lines that provide us with a circumcenter, we created
two Simpson lines, pedal triangles with point P on the circumcenter of the
original triangle. The pedal
points are labeled P1 and P2 and the intersection of the two lines upon which
the pedal triangles lie is labeled S.

We want to
explore the relationship between the arc measure between the two pedal points
and the angle measure of the intersection of the Simpson lines.

As you can
see here the angle measure of the is equal to twice the measure of angle
BSA, which is the opposite angle from the arc. This relationship is maintained
regardless of the translation of the pedal points.