Bill Tankersley's EMT 668 Page
Final Project - Part 3
The assignment for part 3 of the final project is to select and present
a discussion or solution to one exercise/investigation from any of the assignments
that meet each of the specified criteria. The criteria will be discussed
at the end of the writeup.
The investigation I have chosen for part 3 comes from assignment 9 - problem
16:
Animate Pedal Point P about the incircle of triangle ABC. Trace the loci
of the midpoints of the sides. What curves result? Repeat if ABC is a right
triangle.
Click here to download the GSP file for this investigation.
To investigate this problem, I first constructed triangle ABC, its incircle,
and a Pedal triangle RST to triangle ABC from point P in the plane. Here
is a drawing of that construction:
The investigation was to animate Pedal Point P around the incircle and
trace the midpoints (red) of the Pedal Triangle. Here is what we get when
ABC is acute:
Quite interesting!! We see that there are three ellipses formed by animating
pedal point P. Now, let's see what happens when ABC is a right triangle:
Two ellipses and a circle are formed when triangle ABC is a right triangle.
Now, here's a sketch when ABC is obtuse:
We are back to the three ellipses again, although they don't quite connect
as when ABC was an acute triangle.
Discussion of Criteria for Part 3
One of my goals throughout this course was to gain proficiency in using
various computer programs, especially GSP, to demonstrate theorems, relationships,
investigations, etc., to students so that they can investigate these ideas
for themselves. I felt that this investigation fully embodied this goal.
This investigation opened up new elements of mathematics for me for several
reasons. First, I had never heard of a Pedal Triangle or Pedal Point until
this course. (Perhaps I should have heard of these terms before, but I had
not.) So, not only did I learn new terminology and relationships, but I
also gained quite a bit of proficiency using GSP, especially the animate
feature. I had done a few animations prior to this course, but I feel much
more comfortable and proficient with the animate feature after having done
several investigations such as this one. I also feel that this investigation
shows my best work in that the investigation is very thorough. I checked
many different triangles before making the above claims. This investigation
represents one that brings satisfaction in knowing that I am much more capable
now of using software to investigate mathematical ideas.
End of Part 3