How smart were those ancient geometers?
by rrusczyk, Jun 17, 2006, 3:21 PM
I was just looking through the book Journey Through Genius for some interesting bits to add to the Intro Geometry book, and I came across Heron's proof of his formula for the area of a triangle. I suppose most of my readers know this formula: given a triangle with sides of length a, b, c and semiperimeter s, its area is 
. I would hope that many of my readers know how to prove this formula (but am guessing there are at least a few who don't, but should 
). What I'm guessing there are very few of, is readers who could prove it as Heron did. Give it a try. Keep in mind that the geometers of Heron's time did not have algebra or trig. Therefore, we're not talking about using the law of cosines, or about building right triangles and lots of algebra. We're talking about an ingenious construction, followed by a couple simple ratio manipulations.
. I would hope that many of my readers know how to prove this formula (but am guessing there are at least a few who don't, but should ). What I'm guessing there are very few of, is readers who could prove it as Heron did. Give it a try. Keep in mind that the geometers of Heron's time did not have algebra or trig. Therefore, we're not talking about using the law of cosines, or about building right triangles and lots of algebra. We're talking about an ingenious construction, followed by a couple simple ratio manipulations.
February 2012
January 2012