In this assignment, we will explore pedal triangles and some of their properties. Given a triangle ABC, there is a unique pedal triangle for any point in the plane. To form a pedal triangle, find the intersections of the lines containing the sides of triangle ABC with the lines perpendicular to these through the point.
For a GSP script to perform this click here.
Let's explore the pedal triangles of a few interesting points. Let's begin with the circumcenter of ABC.
The circumcenter of triangle ABC lies at the intersection of the perpendicular bisectors of the sides. Clearly, the intersections of these bisectors with the sides give the vertices of the pedal triangle since they are perpendicular to the sides and pass through the point.
Since the perpendicular bisectors are indeed bisectors, these vertices of the pedal triangle are the midpoints of the sides of ABC. Hence, the pedal triangle is the medial triangle of triangle ABC. This is true whether or not the circumcenter is inside or outside ABC.
Now let's explore the pedal triangle of the incenter.
Note that the incenter is equidistant from each side. To find this distance we would measure along the perpendicular from the incenter to each side. Thus, the distance from the incenter to each vertex of its pedal triangle is the same. Also, the incenter is the center of the incircle, which touches each side of ABC. It follows that the pedal triangle for the incenter must be inscribed in the incircle.
What about the orthocenter, which is the intersection of the altitudes?
Clearly, the pedal triangle is the orthic triangle of ABC since the altitudes are perpendicular to the sides.
You can explore the pedal triangles of the centroid and the center of the nine-point circle here.
Now let's consider a point on the side of ABC. If that point lies on the side, then it is the intersection of the perpendicular with the side, and it becomes a vertex of its own pedal triangle.
If the pedal point is a vertex of ABC, then it is the intersection of the perpendiculars and the two sides that it connects. Thus, the pedal triangle is reduced to the altitude from this vertex to the opposite side.
Can you find other points whose pedal triangles are segments? These segments are called Simpson Lines.
It turns out that any point on the circumcircle will give a Simpson Line.
To explore this click here.
Now let's explore what happens when we trace the midpoints of the sides of the pedal triangle and move the pedal point around a circle. Let's begin with a circle centered at the circumcenter of ABC, but larger than the circumcircle.
Each of the loci seem to be a ellipse encompassing the circumcenter and a vertex of the original triangle.
What if we move the pedal point along the circumcenter?
It appears that each locus is an ellipse that contains one of the vertices of the original triangle.
What happens when we use a circle with the same center but smaller than the circumcircle?
We still get an ellipse for each. Now they encompass the circumcenter and are contained inside the original triangle.
To explore these animations, click here.
If we trace the entire line containing the Simpson Line as the pedal point moves around the circumcircle we get a picture such as follows:
Click here for the GSP file.
Finally, let's see what happens when we take a point on the circumcircle and find its pedal triangle and the segment from this point to the orthocenter of the original triangle. Exploration with GSP shows that the segment to the orthocenter is bisected by the Simpson Line. Explore this here.
Can you prove this? Can you prove that the pedal triangle for points on the circumcircle is indeed a line? What other interesting things can you find involving pedal points?