Let the $N$-gon have side lengths $x$. Going around the polygon in clockwise order, let the extended segments to the left of each side be of length $a_i$ and those to the right be $b_i$. By power of a point, $a_k(x+b_k)=b_{k-1}(x+a_{k-1})$ for each $k$. Adding up these equations for all $k$, we have $x\sum a_i + \sum a_ib_i=x\sum b_i +\sum a_ib_i \iff \sum a_i = \sum b_i$. Colouring all of the $a_i$ in red and all of the $b_i$ in blue gives the desired.