On Thu, Aug 14, 2008 at 12:40 PM, Dan Asimov <dasimov@earthlink.net> wrote:
a) For a point outside the circle, its two tangent segments have lengths that total more than the circle's arclength between them. This amounts to showing that
tan(theta) > theta
for 0 < theta < pi/2. I believe Archimedes was capable of verifying this inequality even without calculus and modern terminology for trig functions.
I think that's the hard part -- how do you show that? How did Archimedes do it? (If nobody knows, once I finish unpacking some boxes I should be able to find my copy of his complete works and look it up myself) I agree that it's trivial for the inscribed circle, and that Archimedes knowing this for the circle is far from proving Cauchy's general statement. --Joshua Zucker