Pedal Triangles

Pedal Triangles are formed by constructing a triangle ABC and a point P anywhere on the plane and then constructing the perpendiculars between P and all sides of the triangle(In this case I had to extend the sides). The resulting triangle is called the Pedal triangle(Yellow triangle named RST).

Take a look at this script tool to create your own.

What if Pedal point P is the centroid of triangle ABC?

Point P is not in the interior of triangle ABC. The dashed green lines are the medians of triangle ABC and their point of concurrency is the centroid. When I line up pedal point P with the centroid, the triangle is split into 4 triangles. However, they are not congruent since points S and R do not fall on the midpoints of the sides.

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

Notice again that the pedal point P is in the interior of the triangle. Also notice that it is the point of concurrency of the angle bisectors of the triangle ABC. Again, it appears that there is no concrete relationship between the incenter and pedal point P.

Here we will see how the circumcenter relates to pedal point P.

There is a relationship here because when pedal point P is lined up with the circumcenter, the pedal triangle becomes the medial triangle and divides triangle ABC into 4 congruent triangles.

Here is the construction of the orthocenter.

When pedal point P is joined with the orthocenter, the perpendiculars line up with the altitudes. However, the 4 triangles are not congruent.

There does not seem to be a relationship between pedal point P and the center of the nine point circle. Check out the drawing below.

When pedal point P is dragged to the side of triangle ABC, it lines up with the perpendicular intersection of the triangle. Look below!

These diagrams are the result when pedal point P is lined up with the vertices of triangle ABC.

Shown here is the circumcircle of triangle ABC. Also notice that when pedal point P is on the circumcircle the triangle becomes a line. This is called the Simson line.