1. 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 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 for a script tool for the general construction of a pedal triangle to triangle ABC where P is any point in the plane of ABC.

10. Locate the midpoints of the sides of the
Pedal Triangle (X, Y, and Z). Construct a circle with center at
the circumcenter (K) of triangle ABC such that the radius is __larger__
than the radius of the circumcircle.

Trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle you have constructed.

X = red

Y = green

Z = purple

As you can see, the paths of the midpoints as P animates around the constructed circle are all ellipses. It seems to appear as though A, B, and C are foci of these ellipses, but after further investigation, you can see that they are not. It is true though that A will always be inside the green ellipse, B will always be inside the purple ellipse, and C will always be inside of the red ellipse.

Click here for the animation with GSP.