What If?

Let triangle ABC be any triangle. If P is any point in the plane, then
the triangle formed by constructing perpendiculars to the sides of ABC (extended
if necessary) from P where points R, S, and T are the intersections is called
the **Pedal Triangle**.

Triangle RST is the **Pedal Triangle** for **Pedal Point **P.

Click **here **for a GSP script to construct
a pedal triangle where P is any point in the plane of ABC.

Click **here** for a GSP animation for
various locations of point P.

Next we want to investigate different scenarios for point P depending on the location of P.

Recall the

Click **here** for a GSP animation
for various shapes of triangle ABC.

What did you notice?

Recall the

The incenter is the center of the

First, construct the incenter and the incircle for triangle ABC.

What do you expect when the incenter P becomes the pedal point for triangle
ABC?

The vertices of the Pedal Triangle RST lie on the incircle of triangle
ABC. **WHY**?

The pedal point P is constructed by dropping perpendicular lines from P
to each side of triangle ABC. Then the Pedal Triangle is formed. Its vertices,
R, S, and T are the points of intersection with each side of triangle ABC.
When P is the incenter, the incircle is also constructed by dropping perpendicular
lines from P to each side of the triangle ABC. Hence, when P plays both
roles, the incenter and the pedal point of triangle ABC, the construction
is the same for the incircle and the pedal triangle, RST.

Does this hold for various shapes of triangle ABC?

Click **here** for a GSP animation to
investigate.

Recall the

The **orthic triangle** is the triangle connecting the feet of the
altitudes.

Consider what happens when P is the orthocenter and the pedal point for
triangle ABC.

As expected, the orthic triangle is also the pedal triangle RST.

What if the orthocenter is **outside **of triangle ABC? Does this conjecture
still hold true?

YES.

Click **here** for a GSP animation
for various shapes of triangle ABC and its effect on P and triangle RST.

Recall the

From the construction of the circumcenter, we can construct the **medial
triangle**. The medial triangle is the triangle formed by connecting the
three midpoints of the sides of the triangle.

What if we construct the pedal triangle using the circumcenter as the
pedal point?

The pedal triangle RST is also the orthic triangle for triangle ABC.
This can be examined by investigating the construction of the pedal triangle
RST.

When P is outside of triangle ABC, is the pedal triangle still the orthic
triangle?

Click **here** for a GSP animation
for changes in pedal point P and pedal triangle RST as triangle ABC changes
shape and size.

Recall the

Next construct the pedal triangle RST using the center, P, of the nine
point circle for triangle ABC as the pedal point.

Notice the Pedal Triangle RST will lie on the nine point circle when
the circumcenter, CC, orthocenter, H, and the pedal point, P, all are the
same point; that is, when the circumcenter and the orthocenter become the
center of the nine point circle also known as P.

Here are a few examples:

**What if** ... pedal point P is on the side of triangle ABC?

Begin by constructing triangle ABC with point P on the side of triangle
ABC.

Next construct the pedal triangle RST using P as the pedal point where
P is on the triangle ABC.

What do you discover?

When a pedal point, P, is located on the side of triangle ABC, P will always
be a vertex on the pedal triangle RST.

Construct triangle ABC with pedal point P as vertex A.

This construction was

Think about this...

Given triangle ABC with pedal point P as vertex A.

Next, follow the steps to constructing the pedal triangle RST.

First, construct the perpendicular from point P to side BC of triangle ABC.

Next, construct the perpendicular from point P to sideAB of triangle
ABC.

You notice something very interesting. This step CANNOT be done. It is not
possible.

Try this...

Try constructing the perpendicular from point P to side AC of triangle ABC.
This also is not possible.

Therefore, when the pedal point of a triangle is also a vertex of the triangle,
the pedal triangle cannot be constructed.

This activity was the most interesting and insightful of all the GSP write-ups. This activity connected all of the geometric concepts that we have studied thus far. It was an ideal experience that tied together everything that we have been working on. To complete this write-up, an understanding of all the basic geometrical concepts such as centroid, incenter, incircle, orthocenter, orthic triangle, circumcenter, circumcircle, and the nine-point circle was necessary. This would be a great geometry activity for an overall review and summation of the main geometrical concepts previously mentioned. The connection between the nine-point circle and the pedal point was my favorite.