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.

 

 

Ceva’s Theorem

 

Menelaus’ Theorem

 

Desargues’ Theorem

 

Pappus’ Theorem