Pedals Triangles For All

By:

Alex Moore

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!

1) 2)

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!