__What is
a pedal triangle?__

Given a point P which can be any point in the plane and any triangle, the pedal triangle of P is the triangle whose vertices are the feet of the perpendiculars from P to the side lines.

Click here to go to Math World's website and learn more about pedal triangles.

Let's do a construction of a pedal triangle. First, I am going to take a triangle ABC and pick any point P in the plane. Then, I will construct the perpendicular lines through the point P to each side of the triangle ABC.

Notice the triangle in blue is the triangle ABC and the triangle in black in the triangle RST which is the pedal triangle for the pedal point P.

The yellow lines form the sides of the triangle ABC and the green lines are the perpendicular lines that go through the pedal point P to each side of the triangle ABC.

Click here to for a script tool of the pedal triangle above where you can play around with the triangle.

*Notice* how the pedal triangle
is located all the way inside the triangle ABC when the **pedal
point P is the centroid of the triangle ABC.**

*Notice*, now that the pedal
triangle is located once again inside the triangle ABC when the
**pedal point P is the incenter of triangle ABC. **The pedal
triangles when P is the centroid and the incenter do look almost
the same.

*Let's* look at the pedal triangle
when the pedal point P is the **orthocenter of triangle ABC**.

*Here* is an interesting observation.
When the **pedal point is the orthocenter of triangle**, the
pedal triangle and the orthic triangle are one and the same. This
makes sense when you recall the definition of the orthocenter.
It is the common intersection of the three lines containing the
altitudes. An **altitude** is just a perpendicular segment
from a vertex to the line of the opposite side. Remember, the
definition of a pedal triangle is one whose vertices are the feet of the
perpendiculars from P to the side lines. These definitions form
the same triangle when P and the orthocenter are the same points.

*Now*, let's see what the
pedal triangle looks like when** P is the Circumcenter of the
triangle ABC. **Here, since P is the circumcenter, it is the
center of the Circumcircle or the circumscribed circle of the
triangle.

Notice how the vertices of the pedal triangle lie at exactly the point of intersection where the perpendicular lines that form the circumcenter intersect each side of the triangle.

This is because the circumcenter is found by the intersection of the perpendicular bisectors of a triangle.

*What *happens to the pedal
triangle when the pedal point P is on one of the sides of the
triangle ABC?

*The* pedal triangle still
lies inside the triangle ABC as in all the cases I have explored
so far. Here, the pedal point P serves as one of the pedal triangle
vertices as well. This is the first time this case has happened
thus far.

*Now*, let's look at the
pedal triangle when **P is on one of the vertices of triangle
ABC**. Or, is there a pedal triangle in this case?

There isn't a pedal triangle in this case because two of the perpendicular lines (shown in light blue above) to the sides of the triangle go through the same point. Therefore, you only have two points to construct a triangle with and that can't happen!

The __vertices
of the pedal triangle are colinear and form a degenerate triangle__
when the pedal point lies on one of the vertices of the triangle
that you started with which was ABC in this case. This scenario
was just discussed when I tried to construct a pedal triangle
when the pedal point was one of the vertices of the triangle ABC.
I placed the pedal point on the vertex C and there wasn't a pedal
triangle formed because all of the three vertices of the pedal
triangle were colinear. Two of them were the same point, C and
the other one was the point where the pedal point P perpendicularly
intersected the line segment AB.

*The line
segment formed upon connecting the vertices of the pedal triangle
when all three of the vertices of the pedal triangle are colinear
is known as the*
** SIMSON LINE**.

See the construction below of three Simson lines when the pedal point P is on each of the triangle ABC's vertices.