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Now let AB = d.

So from triangle inequality CA+CB>=d. Thus d/2 <= max(CA,CB)<=d. Now since CD is greater than or equals to both CA and CB, it has to be greater than or equals to d/2 as well. CD has to be less than d as well because if not then CD would have been the diameter. But that is not the case.

Hence d/2<=CD<=d thus if we can find the length of CD we are done with approximating the value of the diameter within a factor of 2 and that we can do by creating BFS tree from C and calculating depth of the tree which is nothing but CD.

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