Using Computers to Explore Geometry

For
our project, we used three geometry software packages to explore some of the
concepts introduced in class. We used
two commercial packages, Cinderella and Geometer’s Sketchpad, and a freeware
program called KSEG, developed by Ilya Baran, a student at MIT. Each program has its own strengths and
weaknesses. Some are better suited for
calculations while others create much clearer illustrations. For our test constructions, we chose Ceva’s,
Menelaus’, Desargues’ and Pappus’ Theorems, since we had just covered them in
class.

The
most successful program for illustrating Desargues’ Theorem was
Cinderella. It allowed us to move
points on the construction more smoothly and naturally, which allows the user
to see how one change can effect the entire construction. In Geometer’s Sketchpad and KSEG, the points
on the construction were constrained in too many places, which made the
construction stiff and unnatural and not as effective at illustrating the
concept.

Unfortunately,
Cinderella did not allow us to calculate ratios in order to illustrate Ceva’s
and Menelaus’ Theorems. For those, we
used Geometer’s sketchpad, which has a ratio function that allows both negative
and positive ratios, which are necessary for proving both theorems.

KSEG
also has a ratio function, but its ratios are always positive, so we could not
use it to prove Ceva’s or Menelaus’ even though it appeared to calculate
Menelaus’ Theorem correctly. We were
able to construct Pappus’ and Desargues’ Theorems easily, though, since they
did not require any calculations.

Both
Cinderella and Geometer’s Sketchpad have the ability to create interactive
webpages, which we have presented here, but to interact with our KSEG
constructions you will have to download the freeware and then the file.

To
download the Windows version of KSEG, click on the link below or enter http://www.mit.edu/~ibaran/kseg.zip
into your
browser's address bar. When the dialog box appears, click on "save
file", and create a folder called kseg to save it in (it may take a while
to download on slower connections). Then use an unzipping utility such as
Winzip to decompress the files into the kseg folder you just created, then
click on kseg.exe to start the program.
You will have to open the file from within KSEG (for some reason, even
if you associate KSEG with that file type in Windows, you still can’t just
double click on it to open it in KSEG.)

Below
are links to the pages for each theorem.
Each page contains a statement of the theorem and screenshots of the
constructions. Click on the screenshots
to go to the interactive webpages (or to download the file, in the case of
KSEG). The interactive webpages may
take a while to load the first time (especially over slower connections), but
once you have visited them, they load much faster. These pages are best viewed with Internet Explorer.