**Assignment #9**
**Nicole Mosteller**
**EMAT 6680**

**Instructions:** Let **DABC** be any triangle. Then if **P** is any point in the plane, then the triangle formed by
constructing perpendiculars to the sides of **DABC** locate three points **R**, **S**, and **T** that are the intersections.
**DRST** is the pedal triangle for pedal point **P**.
**Script 1:** Constructs **DRST** given **DABC** and point **P**.

**When P is the circumcenter of DABC...**
In this investigation into varying the location of the pedal point **P**, we see that when the pedal point **P** is
the circumcenter of **DABC**, the ratio of the perimeters of **DABC:DRST = 1:2** and
the ratio of the areas of **DABC:DRST = 1:4**.
__Perimeter Proof__
Since P is the circumcenter of **DABC**,
we know that P is located on the perpendicular bisector of the sides of **DABC.**
By construction, the pedal triangle's vertices must be the midpoints of the sides of **DABC**.
Now, from the midsegment theorem, we know that in a triangle, a segment drawn through the
midpoints of two of the sides is not only parallel to the third side, but also half the length.
With this in mind, we see that
**RS = 1/2 AB.**
**ST = 1/2 AC.**
**TR = 1/2 BC.**
**RS + ST + TR = 1/2 (AB + AC + BC).**
Perimeter of **DRST** = **1/2** Perimeter of **DABC.**
**DABC:DRST = 1:2.**

__Area Proof__
Recall from the perimeter proof, the sides of the pedal triangle **DRST** are 1/2 the sides of the original triangle **DABC**,
and the vertices of the pedal triangle are the midpoints of the original triangle.
By SSS, we see four congruent triangles, and so the
Area of **DRST** = 1/4 Area of **DABC.**
**Area of DRST:Area of DABC = 1:4.**

**When the Simpson Line appears ...**
As suggested in Assignment 9, question 10 and 11, the midpoints of the sides of a pedal triangle were located, and
their loci were investigated as P was animated around a circle larger than the circumcircle.
Following the instructions, I noticed that the loci of the midpoints are ellipses.
**Click Here!** to see on this animation on GSP.
Next on the list was to let the pedal point P animate around the circumcircle of the original triangle.
Immediately, the vertices of the pedal triangle become collinear. This segment is the Simpson Line.
To see the Simpson Line in Action,
**Click Here!** for an animation on GSP.

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