Workshop on Geometry Software

Math 241
Spring 2015
Professor: Mike O'Sullivan

Week 1: Euclidean Geometry

First Day:

We constructed the circumcircle of a triangle using the perpendicular bisectors of the sides. We discussed why the perpendicular bisectors are coaxial (meet in a single point). We then talked about the angle bisectors of a triangle and why they are coaxial. Students then constructed the incircle of a triangle, which meets each side of the triangle in a single point.


  1. Pepare a Geogebra worksheet with the construction of the circumcircle of a triangle. Use color, highlighting, and text so that your worksheet would be useful to a student. Give some explanation for why the construction works. This doesn't have to be a formal proof, but it should be help illuminate the key aspects of the proof.
  2. Make a similar worksheet for the incircle.
  3. The medians of a triangle meet in the centroid and the altitudes meet in the orthocenter. Make a worksheet that has the circumcenter, incenter, orthocenter, and centroid, See if you can make it not look messy! (Hide some things, use color, ...) Look up the Euler line and make an observation.

Second Day:

We constructed the midpoint quadrilateral for an arbitrary quadrilateral. We discussed why the midpoint quadrilateral is always a parallelogram. We also constructed a rhombus and we talked about their being 2 degrees of freedom in constructing a rhombus: once you specify the length of a side and an angle, the rhombus is determined.

Assignment: Make 4 worksheets, one each for a rectangle, parallelogram, kite, and isoceles trapezoid. For each one

  1. Discuss how many degrees of freedom there are.
  2. Construct the diagonals and observe the properties: (perpendicular? congruent? bisecting each other?).
  3. For two of the figures justify your observation about the diagonals. This need not be a formal proof, but say something about the triangle theorem that is used to show the result.

Upload your worksheets to GeoGebra and put them in two books: one for the triangles and one for the quazy quads. Choose "Share with link" and send me the link via email. (

Due: Monday 2/9 at 5:00 am.

Comments: Here are some observations that should be generally useful to all of you for future assignments and revision of this one for the final portfolio.

  1. Pay attention to spelling, grammar and readability! If I have to struggle to read it, that's bad.
  2. Use proper terminology: triangles have vertices and edges; segments are finite, lines are infinite, rays are infinite in one direction. Use the word congruent (not equivalent). Be sure that the vertices of two congruent figures are listed in the correct order, so that corresponding parts correspond in the lists.
  3. Remember that your audience is your students. You should present so that they can understand and learn. (The goal is not to do the minimal amount to show me that you understand something.)
  4. If I move points, the figure should still remain valid.
  5. Explain your constructions so that someone can follow them and get the correct figure. They are like recipes, (or furniture assembly instructions).
  6. There are often several ways to do a construction, so I don't expect all of you to do the exact same thing. I did expect that you would construct based on the defining property of a figure. For example, a kite has two pairs of adjacent sides congruent. From this we derive that the diagonals are perpendicular, rather than starting with the diagonals and then constructing the figure. I was lenient about this.