& Special Pedal Triangles

This paper (a sort of smorgasbord of observations and conjectures) attempts to address these four questions:

**I. Definition. **Given a triangle, what is a Pedal Point
and its corresponding Pedal Triangle?

**II. Special well-behaved cases.**
What special Pedal Triangle results when the Pedal Point is a
special point (i.e. a center) of the given triangle?

**III.** **Loci. **What
are some interesting loci of vertices of a Pedal Triangle when
the Pedal Point is a special point (not necessarily a center)
of the given triangle?

**IV. A Special Investigation.**
What happens if a Pedal Point is animated around the incircle
of the given triangle and the midpoints of the sides of the Pedal
Triangle are traced? What if instead the excircle is used?

**I. Given a triangle, what is a Pedal Point
and its corresponding Pedal Triangle?**

Given triangle ABC, and let **P**
be any point in the plane. Construct perpendiculars from **P** to the sides of triangle ABC, extending
the sides as necessary. (Notice that *lines* AB, BC, and
AC have been drawn below in the diagram on the right.)

Connect the three intersection points **R**,
**S**, and **T**
to form a triangle, which is called the **Pedal Triangle**
for **Pedal Point P**.

**II. What special
Pedal Triangle results when the Pedal Point is a special point
(i.e. a center) of the given triangle?**

(1) When the Pedal Point **P**
is the **circumcenter** of the given triangle ABC, the Pedal
Triangle is the medial triangle. The reason for this special result
is that AB, AC, and BC are chords of circle **P**
when **P** is the circumcenter.
If a segment drawn from the center to a chord is perpendicular
to the chord, it will bisect the chord. Hence **R**,
**S**, and **T**
must be midpoints of the sides of triangle ABC and so form the
medial triangle of triangle ABC.

Notice that since the medial triangle always remains inside
triangle ABC, all three vertices of the Pedal Triangle will always
remain on the sides of triangle ABC--which is unusual (see diagram
above in the **Definition** section, where two of the vertices
of the Pedal Triangle do not lie on segments AB, BC, or AC)!

(2) When the Pedal Point **P**
is the **incenter** of the given triangle ABC, the Pedal Triangle
is the triangle whose circumcircle is the incircle of triangle
ABC. This result occurs because **P**
is the incenter, and so distances from **P**
that are perpendicular to the sides of triangle ABC must be radii
of circle **P**. Thus, **R**, **S**,
and **T** must lie on circle P.

Notice that since the incircle always remains inside triangle ABC, all three vertices of the Pedal Triangle will always remain on the sides of triangle ABC in this situation as well.

**III. What are
some interesting loci of vertices of a Pedal Triangle when the
Pedal Point is a special point (not necessarily a center) of the
given triangle?**

The following loci are best seen as individual animated GSP sketches, rather than static images (although some images are included below.)

(1) When the Pedal Point **P**
is the **centroid** of the given triangle ABC, at least two
points of the Pedal Triangle appear to remain on triangle ABC
at all times.

In particular, when point B is animated around circle C and
point **T** of the pedal triangle
is traced, it appears that the locus of **T**
is a limacon. **Download this GSP file**
to explore the situation and to determine when the locus will
be a cardioid.

(2) When the Pedal Point **P**
is the **orthocenter** of the given triangle ABC, the Pedal
Triangle is the same as the Orthic Triangle. This result is not
a big surprise, because the lines containing PT, PR, and PS are
all perpendicular to the lines containing the sides of triangle
ABC. So when AP, BP, and CP are all contained on the altitudes
of triangle ABC, points **R**,
**S**, and **T**
are identical to the feet of the altitudes and thus triangle **RST** is the orthic triangle.

When point A is animated around an (arbitrary) circle D and
points **R** and **S**
are traced, the loci appear to be arcs of the same circle, and
sometimes a complete circle. **Download
this GSP file** to investigate when the loci of each point
will be a complete circle. The locus of **S**
is shown below (the image is not faulty--the locus is actually
a major arc and not a full circle.)

(3) When the Pedal Point **P**
is the **on a side** of the given triangle ABC (**P**
coincides with **R** below), interesting
loci result from tracing point **S**
as A is animated around circle D as shown. Notice that in some
cases a sort of "sickle cell" shape appears, while in
other cases, we see a what seems to be a limacon again! **Download this GSP file** to experiment
with this situation.

**IV. What happens
if a Pedal Point is animated around the incircle of the given
triangle and the midpoints of the sides of the Pedal Triangle
are traced? What if instead the excircle is used?**

If Pedal Point **P **is animated
around the incircle of triangle ABC (circle **D**
below), the loci of all three midpoints (**G**,
**H**, and **I**
below) of the Pedal Triangle **RST**
appear to be ellipses. Even more seems to be true: the axes of
the ellipses fall on the angles bisectors of triangle ABC. In
fact, even MORE is true, if one of the angles of triangle ABC
is right.

**Download this GSP file** to
view the animation, to determine whether the major axes of the
ellipses always lie on the angle bisectors, and to see what happens
when triangle ABC becomes right.

If instead **P** is animated
around the excircle, similar results occur. In the diagram below,
**K**, **L**,
and **M** are midpoints of the
sides of Pedal Triangle **RST**
and **P** will move around excircle
**E**.

If you wish, **download one more GSP
file** to explore. (Note that in the file, all three excircles
are shown.)

Although I am far from any sort of explanation or proof of these conjectures, I have determined a few things:

- While an ellipse can be defined as the set of points equidistant
from a point inside a circle and a point on a circle, such a
circle must contain the ellipse, and so cannot be the incircle
or excircle above (since the loci intersect these circles.)

- The circle that circumscribes each ellipse is tangent to
two of the sides of triangle ABC. (That is, the center of this
circle, which is also the center of the ellipse, lies on an angle
bisector of triangle ABC--not terribly surprising.)

- In the case of right triangle ABC,
**P**and the right angle lie on a rectangle containing two other vertices of the Pedal Triangle (**R**,**S**, or**T**.)

That's all for now! Have fun exploring.