Let ABC be an arbitrary triangle and pick any point P, in the plane, then the triangle formed by the intersections of the perpendiculars through point P and the sides of ABC locate three points that make up the Pedal Triangle.
Now we are going to investigate this triangle a little more... (GSP Tool for Pedal Triangle)
Since P can be any point inside ABC, what do you think would happen in the incenter was the point P?
We can recall that the incenter of a triangle is the point of intersection of the three angle biscetors of the triangle. Here is a picture:
Is there anything interesting going on? YES there is. The incircle formed from the original triangle is the circumcircle for the pedal triangle formed. This works for all acute triangles as well.
This is true because to find the radius of the incircle a perpendicular is dropped from the incenter to a side of the circle and the intersection point to the incenter is the radius. That same point is also one of the vertices of the pedal triangle so we have this pretty construction.
Therefore, when the pedal point of triangle ABC is the incenter of that same triangle then it becomes the circmcenter of the pedal triangle!
Does something just as cool happen when the pedal point is the circumcenter of the triangle?
Remember that the circumcenter is formed by the intersection of the perpendicular bisectors of each side of the original triangle.
That means that this pedal triangle is also the medial triangle since the vertices of the pedal triangle are at the midpoints of the sides of the original triangle. And that is the definition of the medial triangle. So, now we have found to cool things about these pedal triangles.
Let's see if we can find some more interesting things by looking at the orthocenter as the pedal point.
First lets remember that the orthocenter is found by the intersection of the three altitudes of the triangle. Once constructed it is obvious that the orthocenter is also the incenter of the pedal triangle and the incirlce fits within this triangle.
We already showed that the incenter of the original triangle is the circumcenter of the pedal triangle, the circumcenter forms the medial triangle as the pedal triangle, and the orthocenter of the triangle is the incenter of the pedal triangle. If we do this with the original, then its pedal, and its pedal and so forth we get:
From the previous statements we can see that P is the pedal point of the green triangle which is also the incenter, the circumcenter of the red triangle, the orthocenter of the blue triangle and the incenter of the orange triangle. This pattern should continue so that the pedal point of the triangle is the incenter of every third triangle. And this pattern does continue. So we can see how if we pick a pedal point that is already another point in the circle we can get this pretty design.