What is a pedal triangle? Good question, actually it is the same one I asked at the beginning of this investigation. First before you can begin creating a pedal triangle you must have a triangle and a random point P somewhere on the plane. The pedal triangle is formed when you drop perpendicular lines from point P to the sides of the given triangle, and the connecting of those intersection points become the vertices of the pedal triangle, as shown below.

Pedal point interior	Pedal point exterior

A number of relationships come from the study of the pedal triangle. I have chosen to focus on a few specific cases:

• the pedal point and its position in relation to the pedal triangle
• the pedal point on a excircle of the original triangle
• the use of area to find the Simpson Line

PEDAL POINT POSITION & THE PEDAL TRIANGLE

For sake of ease of discussion the Point P is the Pedal Point, triangle ABC is the "O"original triangle, and the red triangle is the "P"edal triangle.

(Left) Pedal Point outside "O" triangle - When the pedal point is exterior to the "O" triangle, the pedal point will always be outside of the boundary of the "P" triangle. This is always true because one of the veritices of the "P" triangle will always be on the side of the "O" triangle that point P is closest to.

(Right) Pedal Point on a side extension of the "O" triangle - When the pedal point is on a side or a side extension of the "O" triangle, the pedal point becomes one of the vertices of the "P" triangle. The occurs because the perpendicular from the pedal point to the "O" triangle is itself. This occurs on any side or side extension.

(Left) Pedal Point is inside the "O" triangle - When the pedal point is inside the "O" triangle, the perpendicular lines from that point create a "P" triangle that is formed around the pedal point. Once the pedal point enters the "O" triangle it also enters the "P" triangle.

(NOW YOU TRY) I welcome you to manipulate the pedal point below, just grab it and move it around. Do you find what I found?

THE EXCIRCLE AND THE AREA OF THE PEDAL TRIANGLE

Moving the pedal point around the excircle we find a few interesting mathematical discoveries. First we find that the pedal point becomes a vertex of the pedal triangle three times. This occurs at the three points of tangency of the excircle to the side extensions of the "O" triangle. This is simply a confirmation of a previously stated fact that the pedal point becomes a vertex of the "P" triangle when it is on a side of the "O" triangle. Second when we investigate area of the "P" triangle we see find the Simpson Line.

Area of the Pedal Triangle - As the pedal point is moved around the excircle we find two times that the area of the "P" triangle is equal to zero.

Hopefully my hint helped you find that as the pedal point moves around the excircle, the "P" triangle has an area of zero at two exact points.These points are the intersection points between the excircle and the circumcircle of the "O" triangle. If we were to move the pedal point around the circumcircle we would find that the area of the triangle is equal to zero at every point. The degenerate line formed by this "P" triangle is called THE SIMPSON LINE.

I will leave the investigation of when is the area is a maximum up to you to discover. I will guide you in asking, what do you notice about a vertices of the "P" triangle and the "O" triangle when the area is a maximum?