Plenárias

GETTING EULER'S LINE TO RELAX
Paul Goldenberg
EDC Newton - USA

Often, in school investigations of a theorem (or a mathematical phenomenon), the focus is only on the "conclusion" part, the consequences that follow from the "hypothesis," a stated set of conditions. Students are, for example, given the hypothesis and asked to prove the conclusion. Mathematicians also work in the reverse direction, examining the hypothesis of a theorem to see whether all of its elements are essential to the conclusion, or to see how the conclusion generalizes as the hypothesis is weakened.

To experience mathematics as a mathematician does, students must be able to workin both directions. This paper will present an exploration perfect for students using Cabri. Using common-sense notions -- no fancy knowledge, but support from Cabri and some sophistication -- students can investigate the hypothesis behind the Euler Line, the collinearity of three particular "centers" of a triangle. As they apply and practice some powerful mathematical ways of thinking, the discoveries they encounter on the way are truly astonishing.