More GeoGebra Examples:
Problems and Explorations:
The first example, here, is a geometric "fact" I stumbled on while exploring possible max-min problems.
Given any triangle, ABC, with sides a, b, c: Select a point, D, on side c, and a point E, on side a. Connect A to E, with midpoint, H. Connect C to D, with midpoint, I. Drop Perpendiculars from D and E to line CA (side b, extended to form a line.)
Show that the area of triangle HIB equals the area of quadrilateral HIGF.
I have "fact" in quotes, for now, because I have yet to come up with a geometric proof of it, although I'm sure it's not deep. It's probably a variation on the standard construction of a triangle with area equal to a quadrilateral, but I'm not seeing it. I did come up with a coordinate proof, I think, but it was very messy.
Please e-mail me, if you come up with a proof, and I'll post it here, with my thanks and your credit.
*** Update: 24/04/09: Michael Tzoumas was smart enough to solve this, and kind enough to e-mail me the solution. I've posted it below the applet (click on the applet link.)
The next example is a simple max-min exploration that I found in April's Augarithms (Ken Kaminsky's wonderful newsletter from Augsburg U.'s math department (http://www.augsburg.edu/math). It's taken from Bradley University's math department's problem of the week page, http://hilltop.bradley.edu/~delgado/potw/potw.html. It provides a nice example where figuring out how to construct the applet, also shows you how to solve the problem.
Last year, at MCTM/MinnMATYC - Duluth '07, I gave a talk about projects where the students do more of the leg-work in creating real-life examples. One more directed version of this is the "maximize the kicking angle" problem, which I thought I'd invented during a heated discussion one Thanksgiving back when Nebraska and Colorado both had good football teams. Turns out, a lot of other people invented the problem before I did. Anyway, I made an applet to explore the relationship between ball placement, hash mark widths, goalpost widths, and maximal angles. The students are supposed to find the actual field dimensions themselves:
Still more to come....