Pedals Triangles For All
In our next investigation we move to pedal triangles. What are pedal triangles you ask? Given any triangle ABC, the pedal triangle of ABC at pedal point P is the triangle defined by the intersections of the perpendicular bisectors of the sides (or extensions of the sides) of ABC through the point P. A basic example is shown below. The red triangle is the pedal triangle
The observant reader might notice that P is neither in the interior of ABC and not on the boundary of ABC. Also, what if P was elsewhere outside ABC? What if ABC is a less general triangle, like a right triangle? Or an isosceles triangle? Or an equilateral triangle? Is the pedal triangle always a legit triangle (i.e., nonempty interior)? There are many questions to explore.
Let us first explore with P outside of ABC. What if P were on the upper half plane defined by 1) AB, or 2) AC? Let us see!
As we can see, we have examples of the pedal triangle being both inside and outside the triangle in both of these cases. What if the pedal point is in the interior of the triangle?
We note that if P is an element of the interior of ABC then the pedal triangle will clearly have one vertex on each of the sides of AB. This is obvious by the definition of the pedal triangle. What if we place P on the side, not on a vertex?
Nothing surprising arises here. However, let us consider the case of a right triangle. What if we place P on ABC, not at a vertex, on a right triangle?
We see in the first two instances that one of the sides lies on the sides of ABC. In the last case it appears that the pedal triangle is a right triangle. It is again clear that if P is any point on the hypotenuse of ABC the pedal triangle will be a right triangle. Why is this obvious? By definition. Since two of the sides are perpendicular to two sides of ABC, and those two sides of ABC are perpendicular, our pedal triangle must be a right triangle. Students could continue to explore different scenarios for the pedal triangle of a right triangle, or begin explorations on isosceles triangles! We finish here by beginning an exploration of the degenerate pedal triangle (the three vertices of the pedal triangle are collinear).
As an obvious first guess, we would expect to have a degenerate pedal triangle if P was chosen to be a vertex of the triangle. Why is this clear? If we chose P to be a vertex, then two of the vertices of the pedal triangle are the same point! Since the triangle is made up of segments perpendicular to the sides (or extensions thereof) of ABC, if P rests on one of the sides then P is a vertex of the pedal triangle. So if P is a vertex ABC then P is on two sides of ABC and is therefore two vertices of the pedal triangle. Let us demonstrate.
Since we know that the pedal triangle is degenerate when P is a vertex of ABC we naturally ask: what if P is on the circumcircle of ABC? Since the vertices of ABC are on the circumcircle, why would we not ask this question next? Let us explore!
The pictures here certainly seem to suggest that this interesting triangle is degenerate when P lies on the circumcircle, and as it turns out this is the case. What if we now trace the degenerate triangle (this is known as SimpsonŐs line)?
In my opinion this is not very interesting. Even though this example is not fancy, student can experiment with the trace for different types of triangles, like equilateral, or right triangles, or isosceles triangles. There is a whole left to explore!