Pedal Triangle Exploration
by
Laura Kimbel
The Pedal Triangle is constructed by first drawing a triangle and picking an arbitrary point. This point will be our "Pedal Point." Next, we construct the lines that are perpendicular to each side of the triangle but also go through the pedal point. Next, locate the points of intersection between our perpendicular lines, and the sides of our triangle. [Keep in mind, this intersection point may be outside of our triangle, and therefore we must construct our triangle with lines, instead of line segments.] These 3 intersection points form our "Pedal Triangle." The pedal triangle can lie completely within our original triangle, it can cross our original triangle, or it can lie completely outside of our triangle. This all depends on the type of triangle that our original triangle is, and also where our Pedal Point is located. We will now investigate different locations of our Pedal Point and see how it affects the position of our Pedal Triangle.
What if the Pedal Point is the Centroid?
As, you can see, we have 2 different cases here. First, we have that the green triangle (our original triangle) is accute. When we merge the Pedal Point with the Centroid of an accute triangle, we can see that the Pedal Triangle lies completely within our green triangle. However, when our pedal point lies on the Centroid, that does not always mean our Pedal Triangle lies within our green triangle. In some obtuse triangles, like the second green triangle above, part of the Pedal Triangle may lie outside our original triangle. Even though in both cases, our Pedal Point is in the interior of our triangle, that does not necessarily mean that our Pedal Triangle is.
What if the Pedal Point is the Orthocenter?
Now we have the Pedal Point as the Orthocenter of the triangle and again we look at 2 cases; accute and obtuse. In our accute triangle, the Pedal Point and the entire Pedal Triangle lie within our original triangle. However, in our second case, we can see that the Pedal Point and part of the Pedal Triangle lie outside our original triangle.
What if the Pedal Point is the Circumcenter?
When the Pedal Point lies on the Circumcenter, we can again consider first, an accute triangle, and second, an obtuse triangle. For an accute triangle, our Pedal Point and Pedal Triangle are inside our original. However, for our obtuse triangle, we have the Pedal Point outside the triangle but the entire Pedal Triangle within. This is different from our first 2 investigations.
What if the Pedal Point is the Incenter?
For our last investigation, I also included the Incrircle in my picture. As you can see, the 3 vertices of our Pedal Triangle lie on the incircle. The incenter is constructed from the intersection of the angle bisectors. The Incircle intersects our triangle in 3 places. Those points are located by constructing perpendicular lines through our incenter and the sides of our triangle. When the Pedal Point lies on the incenter, these intersection points coincide with the vertices of our Pedal Triangle.