Using mathematical induction, prove that $${}^m{C_0}{}^n{C_k} + {}^m{C_1}{}^n{C_{k - 1}}\,\,\, + .....{}^m{C_k}{}^n{C_0} = {}^{\left( {m + n} \right)}{C_k},$$
where $$m,\,n,\,k$$ are positive integers, and $${}^p{C_q} = 0$$ for $$p < q.$$

Prove that
$${C_0} - {2^2}{C_1} + {3^2}{C_2}\,\, - \,..... + {\left( { - 1} \right)^n}{\left( {n + 1} \right)^2}{C_n} = 0,\,\,\,\,n > 2,\,\,$$ where $${C_r} = {}^n{C_r}.$$

Let $$ABC$$ be a triangle with $$AB = AC$$. If $$D$$ is the midpoint of $$BC, E$$ is the foot of the perpendicular drawn from $$D$$ to $$AC$$ and $$F$$ the mid-point of $$DE$$, prove that $$AF$$ is perpendicular to $$BE$$.

The line x + 3y = 0 is a diameter of the circle $${x^2} + {y^2} - 6x + 2y = 0\,$$.