I have been tasked with an assignment to come up with a 30 minute lesson in Geometry using a software demonstration (Sketchpad, Cabri, WinGEOM, anything) at a high school/undergraduate level on “something interesting”.
Now, I have a vague idea of what I want to do, probably an interactive investigation of extending the idea of a nine-point-circle to some non-euclidian geometry (I haven’t checked yet, for all I know it doesn’t extend to ANY non-euclidian geometry), but that’s just… BORING.
You guys know of any truly beautiful proofs that I can graphically demonstrate instead? Something that takes 15-25 minutes to describe in detail to a high schooler?
A geometric explanation of the pythagorean theorem?
That might be too basic, depending on what level of math they’re at, but it’s a hugely basic and important fact with a nice visual explanation which I’ve only rarely seen discussed.
A discussion of newton’s method might be interesting, as you have little lines bouncing all over the place, intersecting the axis, etc. This would be particularly interesting if you could come up with some examples in which newton’s method didn’t work and show why it was.
I don’t know if you have the time, but you could grab some of Newton’s proofs from a translation of the Principia. He used geometry for almost everything, including proving that planets (objects undergoing a central force) move in a plane.
Alternatively, you could show that a vector dragged around a globe on some trajectory that returns to the starting point does not necessarily return pointing in the same direction.
Take a browse through Euclid’s Elements. Prop 47 of book I (the Pythagorean theorem) is awesome. He also does a whole lot (in the middle books, I think it starts around book 5) with ratios and proportions, but does it all geometrically.
Given a circle C with center O, a point P on C, and a line l which intersects C at P, l is tangent to C iff l is perpendicular to the radius OP.
It can be done using nothing but high school-level geometry and algebra, it should take about the right time to present, and it’s likely to be the most complicated proof the students have seen to date.