Lines and Points in the Plane

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Grade Vs Difficulty:

This math circle lesson plan investigates several basic results in discrete geometry, specifically the Sylvester-Gallai theorem, the Erdos-de Bruijn theorem, Radon's lemma, Helly's theorem, and Tverberg's theorem.

This plan consists of six sets of questions based on these results. These questions are open-ended and intended to assist students in conjecturing these results on their own, rather than stating the theorems and then having students investigate.

These problems vary in difficulty from moderate to perplexing. Parts of this lesson plan can be used for individual math circles, or the entire plan could be used for a multiple-session sequence of circles. This lesson plan is adapted from one used for two sessions of the Central Kentucky High School Math Circle during Fall of 2010.

The pdf file given below contains questions to pose to students, along with remarks for the session leader. These remarks contain both historical information about the problems and resources for leaders (and students!) to find out more about these types of problems.