Given the regular nonagon below
Prove that in any triangle the following equivalence relation holds:
We are working on the following shape.
Let be the incenter of the triangle. Thus,
meaning that is tangent to the circumcircle of the triangle . Thus,
Given a heptagon of side and diagonals such that ,
Let be the side of the heptagon and be its diagonals respectively. It holds that
Equation comes naturally from Vieta’s formulae since are the roots of the equation . Thus,
Let be a function such that and is differentiable at . Let us set
Evaluate the limit .
Since is differentiable at , there is some such that
and is of course continuous.
Let . There exists such that which in return means that . Hence , for larger than it holds that
On the other hand , the sum is a Riemann sum and converges to .
Let denote one of the Jacobi Theta functions. Prove that
We have successively,
The sum is evaluated as follows. Consider the function
and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus,
and the exercise is complete.