The investigation of the Pedal Triangle associated with a Pedal Point of a triangle

What is a Pedal Triangle?

If P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC locate three points R, S, and T that are the intersections. Triangle RST is the Pedal Triangle for Pedal Point P.

We can notice there is some instances that Pedal triangle is degenerated. Of course, if P is a vertex then Pedal triangle becomes a height of the triangle through the vertex.

Conditions in which the three vertices of the Pedal triangle are colinear

(This line segment is called the Simson Line)

Through an exploararion we can know that if P is inside traiangle then Pedal triangle is always made. And then where of outside is a Pedal triangle degenerated?

Construct a circle with circumcenter of ABC as center and with larger radius than cicumcicle radius. And then observe the Pedal triangle. Let's see the locus of the midpoints of the sides of the Pedal triangle as P is animated around the circle.

Three locuses are ellipses.

What if the circle is circumcircle?

We can explore that Pedal triangle is degenerated if a pedal point is on circumcircle. Click here.

Is there a point on the circumcircle for P that has side AC as its Simson line? AB? BC? Yes.

What is the relationship between two Pedal points and their Simson lines?

Through exploartion(Click here) by GSP, there is relationship of twice angle between the angle made by pedal points and circumcenter and the one made by two Simson lines.