# Investigations
with the Pedal Point, the Pedal Triangle, and the Incircle

### by: Lauren Wright

In this investigation,
we begin with any triangle ABC. Then if P is any point in the
plane then the triangle formed by constructing perpendiculars
to the sides of ABC (extended if necessary) locate three points,
R, S, and T that are the intersections. Triangle RST is the Pedal
Triangle for Pedal Point P.
For example:
Click
here to explore this
construction yourself.

In this exploration, we construct the
incircle of triangle ABC and then choose P to be a point on the
incircle.
Then we construct the
midpoints of the sides of the pedal triangle RST.
Next, we animate the
pedal point P about the incircle of ABC and trace the loci of
the midpoints of the sides.
And we see that the loci
of the midpoints of the sides of the pedal triangle form ellipses
as P is animated along the incircle.
Click
here to explore this
construction yourself.

Now, let's look at what
happens when ABC is a right triangle.
Here we see something
interesting happening - the blue locus
appears to form a perfect circle that is tangent to ABC at two
points.
Click
here to explore this
construction yourself.

Now the question becomes,
why does this happen?
We begin by drawing in
some auxiliary lines.
Notice here that ray
AP always goes through that blue midpoint, which we have now labeled
M.
Now, let's take a look
at that right triangle that has been formed, triangle APT.
It appears that M is
the midpoint of the hypotenuse of triangle APT - we can verify
this with GSP.
Or, we can look at rectangle
ARPT. RT and AP are diagonals of this rectangle and diagonals
of a rectangle always bisect each other. So, M must be the midpoint
of AP.
Now, we can make a triangle
similar to APT by dropping a perpendicular from M to segment AT.
These two triangles will be similar by angle-angle similarity.
Since triangle AMN is
similar to triangle APT, as point P travels around the incircle,
M will also form a perfect circle - a circle that will in fact,
have half the radius of the incircle.

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