Through the incentre $I$ of triangle $ABC$ a straight line is drawn intersecting $AB$ and $BC$ at points $M$ and $N$, respectively, in such a way that the triangle $BMN$ is acute- angled. On the side $AC$ the points $K$ and $L$ are chosen such that $∠ILA = ∠IMB$ and $∠IKC = ∠INB$. Prove that $AC = AM + KL + CN$.
I have no idea how to start.
I can only see triangle IMC' and ILB' are congruent but not triangle IMC' and IKB'. I wonder if my diagram is different from yours.