Pedal Triangle

Jeff Dozier

If we have any triangle ABC, we can construct a pedal triangle with any point, P in that plane. Just place a point P and draw perpendicular lines to the sides of ABC (the sides may need to be extended to lines to find intersection points). These three points of intersection will form what is called the pedal triangle.

Click here to open a GSP file and experiment with the pedal point.

Next, let's construct a circle around triangle ABC which has its center at the circumcenter but has a radius larger than the circumcircle. We can locate the midpoints of the pedal triangle and trace them as P is moved around the constructed circle. As P is moved around the circle, the trace of the midpoints will form three ellipses.

Click here to go to this file and move P around the cirlce.

Now, let's examine what happens when triangle ABC is right or obtuse. When it is right, the midpoints will form two ellipses and the other trace will make a circle. When it is obtuse, we go back to having three ellipses.