### EMAT 6680 Assignment #9

### Pedal Triangles

The purpose of this
assignment is to investigate Pedal triangles. We begin by
constructing any triangle ABC. We construct the pedal triangle by
choosing any point, P, and drawing the perpendicular from point P
to the sides of triangle ABC. Click here to see this sketch in
GSP.

We can manipulate P to investigate the outcome if P is different triangle centers.
First we move P to the incenter of the triangle. The vertices of
the pedal triangle are the intersections of the perpendiculars
from the incenter (in dark blue) and the sides of triangle ABC.
Click here to see the animation and manipulate this sketch.

We can move P to the
orthocenter of triangle ABC. The vertices of the pedal triangle
are the intersections of the altitudes and the sides of triangle
ABC. Click here to see this animation and manipulate this sketch.

If we move point P to
a side of triangle ABC, then the pedal triangle is inscribed in
triangle ABC with point P being one of the vertices. Click here
to see this animation in GSP.

Next we can set the
path of P to be the circumcircle of triangle ABC. We can see that
the pedal triangle degenerates and the Simson Line (in red) is
formed. If we trace the midpoints of the pedal triangle sides as
point P is animated around the circumcircle we can see their
paths. The midpoints follow elliptical paths which go through the
midpoints (Q, R and T) of the sides. Click here to see this
animation in GSP.

We can see from this
sketch that when point P is at one of the vertices of triangle
ABC the Simson Line becomes the altitude passing through that
vertice. If we construct the orthocenter we can make another
observation. We can see from the sketch below that the Simson
Line becomes one of the sides of triangle ABC when the segment connecting point P and the orthocenter of triangle ABC intersects
the sides at the midpoints.

Finally if we animate point P
around the incircle of triangle ABC and trace the midpoints of
the pedal triangle we can see their elliptical paths. Click here
to see the animation in GSP.

If we repeat the
animation of P around the incircle when ABC is a right triangle
one of the midpoints paths becomes a circle.