Withdrawn, accompanied by my (by now customary) red face! WFL On 2/13/10, Joshua Zucker <joshua.zucker@gmail.com> wrote:
On Sat, Feb 13, 2010 at 8:47 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
"The theorem of Bang (1897) states that the lines drawn to the polyhedron vertices of a face of a tetrahedron from the point of contact of the face with the insphere form three angles at the point of contact which are the same three angles in each face."
This is obviously false --- consider the limiting case of a semi-infinite triangular prism with equilateral cross-section: the angles at the finite base are all 2pi/3, whereas at the infinite sides they are pi/2, 3pi/4, 3pi/4.
I don't replicate the result you get here.
I find that the point of contact of the insphere with the semi-infinite sides is s/(2*sqrt(3)) above the base of the prism, thus forming more 30-60-90 triangles and repeating the 2pi/3 angles as Bang's theorem would give, not 45-45-90 triangles as you indicate. So at least one of us has the wrong answer here.
--Joshua Zucker
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun