This is an exploration involving Pedal Triangles. First, a few definitions:
Start with any triangle and any point P in the plane. Construct lines through point P perpendicular to the lines containing the sides of the triangle. Mark the intersection of these lines and connect them. This is called a Pedal Triangle. Point P is a pedal point.
With this basic structure, there are many interesting things to explore--here we will look at just one--an interesting attribute of the midpoints of the sides of the Pedal triangle.
We first construct a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcircle. We then trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around this circle. One such construction is seen below. To explore this construction with different size triangles and circles, you can open a GSP file by clicking here.
