Workshop on Geometry Software

Math 241
Fall 2015
Professor: Mike O'Sullivan

Week 1 Geogebra Tools and Constructions Summary

Week 1: Euclidean Geometry

First Day:

We constructed the perpendicular bisector of a segment. We then 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). Next, we talked about the construction of an angle bisector. The angle bisectors of a triangle are coaxial, and their intersection point, I, is called the incenter of the triangle. Drop a perpendicular from the incenter to a side, and call the intersection point P. The circle centered at the incenter I and with radius IP is the incircle; it meets each side in a single point.


  1. Pepare a Geogebra worksheet with the construction of the angle bisector of angle. List each step in the construction. Explain why the construction works. State the property of points on the angle bisector: they are equidistant from the sides. Use color, highlighting, and text so that your worksheet would be useful to a student.
  2. Make a worksheet for the incircle. List each step in the construction (but now you can use GeoGebra's built in tools for the angle bisector and perpendiculars). Give a brief explaination of why the construction works. Use color, highlighting, and text so that your worksheet would be useful to a student.
  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 tried using many of the construction tools in Geogebra. We constructed a square and a rhombus. A square has one degree of freedom (the length of the sides), and a rhombus has two degrees of freedom. Once you specify the length of a side and an angle, the rhombus is determined.

Here is a summary of the key construction tools in Geogebra, the main theorems from geometry that you should know, and the definitions of special quadrilaterals.

Assignment: Make 3 worksheets, one for a rhombus, and two others for a parallelogram, a kite, or an 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 one 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. Choose "Share with link" and send me the links via email. (

Due: Wednesday 9/9 at 5:00 pm.

Comments: Here are some observations that should be generally useful for all of your assignments.

  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. Target an audience of high schoolr 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. Do the construction based on the 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.