To begin this investigation, we must first construct the Pedal triangle. Begin by constructing triangle ABC. Then, pick a point p that is in the plane of ABC. Locate three points R, S, and T that are the intersection points of the perpendiculars from point P to the sides of ABC. Triangle RST is called the Pedal Triangle for Pedal Point P.
The green triangle is the Pedal Triangle for Pedal Point P. During this investigation, we will construct several Pedal Points and discover the different shapes of the Pedal Triangle. I would like to first try point P on the side of triangle ABC.
Notice, when we move point P to lie on a side of the triangle, the one of the points of the Pedal triangle also lies on the side of triangle ABC. It is obvious why this happens, the construction of the Pedal triangle is the intersection of points from point P to each side of the triangle. So if point P lies on the perimeter of the triangle, then the intersection of the point and perpendicular line will also lie on the perimeter.
What if point P lies on the vertices of triangle ABC?
The Pedal Triangle now forms a line called the Simson Line. The Pedal Point lies on vertex A. In addition, the R and T vertices also lie on vertex A. The Pedal Triangle now forms a line segment from the vertex the Pedal Point to point S. After testing all three cases, I found the Simson Line is formed no matter what vertex the Pedal Point lies on.
Now let's examine some special cases. What if the Pedal Point lies on the centroid (G) or the Incenter (I)? Any predictions?
Figure 1 (Centroid) Figure 2 (Incenter)
Point P lies on the centroid and incenter above. The Pedal Triangle formed lies internal to the original triangle. The Pedals Triangles vertices lie on the sides of the original ABC triangle. The vertices RST do not lie on the intersection point of the angle bisector (figure 2) or the midpoint (figure 1). Therefore, the Pedal Triangle is not the medial triangle. Also, I measured the sides of the Pedal Triangle and I could not find any correlation between the sides. Below is the case where the Pedal Point lies on the orthocenter:
Of course, when point P lies on the Orthocenter H of triangle ABC, the vertices of the Pedal Triangle lie on the lines that form the altitudes of the triangle. Why does this happen? This is due to the construction of the Pedal triangle. At first glance, I thought the construction formed an isosceles triangle, but I measured the sides and it did not form a triangle.
To conclude this investigation, I would like to look at the envelope of the Simson line as the Pedal point is moved along the circumcircle. The animation is below:
Notice, as point P moves towards the vertices of triangle ABC, there seems to be a concentration of lines. What happens if P follows a path of a circle that has a larger radius and smaller radius than the circumcirlce?
Larger Radius Smaller Radius
The concentration of lines for the larger radius seems to be a partial triangle, but the construction is mainly sporadic. The smaller radius is similar to the circumcircle envelope, except the lines are missing around the vertices ABC, because the circle does not cover those endpoints. These "holes" form a U shape like a parabola.
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