**Assignment 8**

For this assignment, I chose to write up problem number nine:

*Let triangle ABC be any triangle. 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.*

*Find all conditions in which the three vertices of the Pedal triangle
are colinear (that is, it is a degenerate triangle). This line segment is called
the Simson Line.*

This picture shows pedal triangle **DEF** to triangle **ABC**
for pedal point **P**. This GSP
file contains a script tool for the general construction of a pedal triangle
to triangle **ABC** where **P** is any point in the
plane of **ABC**. I used this script tool and discovered several
places (the green points) where I could place **P** so that the
pedal triangle would be degenerate:

These points appear to lie on the circumcircle of the triangle.
The following theorem is indeed true: **P lies on the circumcircle of
ABC if and only if the pedal triangle DEF to triangle ABC of point P is degenerate.**
After failing numerous attempts to prove this fact, I could not resist peaking
at this
proof. I had more success at proving the following MUCH simpler facts:

**The pedal triangle to triangle ABC of point P, when P
is the incenter of ABC has vertices DEF, which are the points on triangle ABC
tangent to the incircle.**

By definiton of pedal triangle, **PF** is perpendicular
to **BA**, **PD** is perpendicular to **BC**,
and **PE **is perpendicular to **AC**.

**The pedal triangle to the triangle ABC of point P, when P is the orthocenter
of ABC is the medial triangle of ABC.**

By definition of orthocenter, the line through **P**
and **F** is the perpendicular bisector of **AB**,
the line through **P** and **D** is the perpendicular
bisector of **BC**, and the line through **P **and
**E** is the perpendicular bisector of **AC**.