Write Up # 9

Pedal Triangles

by: Angel R. Abney

Let Triangle ABC be any triangle. then if P is any point
in the plane, the triangle formed by locating three points R, S, and T that
are the intersections of the constructed perpendiculars to the sides of
ABC(extended) is called the **Pedal Triangle** for **Pedal Point**
P.

To construct the Pedal Triangle, draw triangle ABC and point P is any point in the plane (See picture above). Then construct the perpendiculars to sides AB, AC, and BC from the point P. The intersections of the perpendiculars and the sides are points R, S, T. the thick blue triangle below is the Pedal Triangle, and P is the Pedal Point.

To manipulate the construction of the Pedal Triangle, click
on **Pedal script**. You must have GSP in order
to run the script. Select Points ABC and P, then press
play on the script. You can then move the Pedal point, P around to see how
this affects the Pedal triangle.

Notice, that as P is moved to the side of triangle ABC, P becomes one of the vertices of triangle RST (see picture above). If P is moved to line BC, then P is concurrent with S. If P is moved to line AC, P lies on the same point as T, and if P is moved to line AB, P is concurrent with R.

**Observations:**

1. If P is moved to the sides (even extended sides) of
triangle ABC, then P becomes one of the vertices of the Pedal Triangle see
picture above. Click **here** to see proof.

2. If P is inside of triangle ABC and triangle ABC is acute, then the vertices of the Pedal Triangle lie on segments BC, AB, and AC.

3. If P lies on a vertex of triangle ABC, then P coexist
with two vertices of the Pedal triangle as well. Thus, Triangle RST becomes
a segment. In other words, it is a degenerate triangle. This line segment
is called the **Simpson's Line**

4. If P is the **orthocenter** (intersection of altitudes)
of triangle ABC, then R, S, and T lie on the intersection of the altitudes
and the sides of the triangle. If the orthocenter lies outside of triangle
ABC, then two of the points of the Pedal triangle lie on the extended sides
of triangle ABC (see second picture below).

5. If triangle ABC is a right triangle, and P lies on the
side opposite the right angle, then triangle RTS is a right triangle. Click
**here** to see proof.

6. If triangle ABC is a right triangle, then angle RPS is always a right angle and line RP is parallel to line BC.

7. If the Pedal point is the **In-center** (intersection
of the angle bisectors), then points R, S, and T always lie on the non-extended
sides of triangle ABC, since the in-center is always inside of triangle
ABC.

8. It appears that the measure of angle ABC is either equal to the measure of angle RPS, or that angle ABC and angle RPS are supplementary.

9. If P is the **circumcenter** ( intersection of perpendicular
bisectors), and triangle ABC is a right triangle, then P will lie on point
R. Thus, triangle RST will also be a right triangle.

10. If P is the **circumcenter**, triangle RST is inscribed
in triangle ABC (even if P is outside of triangle ABC).

11. It should be obvious that if P is the **circumcenter**,
then points R, S, and T lie on the intersection of the perpendicular bisectors
and segments AB, BC, and AC. Therefore, R, S, and T are always the mid-points
of segments AB, BC, and AC, respectively.

12. If P is the **circumcenter**, then the sides of
the Pedal triangle are parallel to the sides of triangle ABC.

13. If P is the centroid (intersection of the medians) of triangle ABC and triangle ABC is isoceles, then the pedal triangle is also isoceles.