By the secant theorem:

|AC|*|CN| = |BC|*|CM|     (note that CM and CN are the sections outside the circle)



Now notice |AC|/|CM| = c = |BC|/|CN|   which means we have two sides of two triangles with a common ratio of side lengths (here, AC corresponds to CM and BC corresponds to CN).   Note also that angle ACB is common to both triangles and is the included angle, thus by SAS triangles ABC and NMC are similar. 



   

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