AUSTRIAN-POLISH 1980 Problem 9

Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$ . Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$ . Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$ . Prove that $M_1M_2 = N_1N_2$ .