Pedal Triangles

First construct any triangle ABC. Then place a point P in the plane.Construct perpendicular lines from the point P to the sides of the triangle. Mark these intersection points.

Now we will connect the points of intersection to form a new triangle. The point P is the pedal point and the new triangle formed is the pedal triangle.

What if the pedal point is the centroid of the triangle?

In this particular instance the triangle formed with the centroid as the pedal point is inside the triangle, but that is not always the case.

What if the pedal point is the incenter of the triangle?

This forms a pedal triangle very similar to the triangle with the centroid as the pedal point.

What if the pedal point is the circumcenter of the triangle?

This triangle is different from those formed by the incenter and centroid.

What if the pedal point is the orthocenter of the triangle?

Now isn't this interesting. The triangle formed when the orthocenter is the pedal point does remain inside the original triangle.