(C) The bible is God's inspired word, free from error.
(A) Why do you believe that?
(C) Because the bible says so.
Technically, this might not be arguing in a circle, because perhaps the Christian has some independent reason for believing that 2 Timothy 3:16 (which is generally used at least to show that the bible is God's inspired word) is true that would be revealed when asked for, but I'll take this as a clearly circular argument.
I have noticed in the past that atheists sometimes seem to argue in a circle when dismissing miracles out of hand.
(A) There is no God.
(C) But what do you make of the miracles that Jesus did, which seem to be historically recorded for us? That's why I believe in God.
(A) I neither know nor care about the historical evidence for these miracles, because miracles cannot occur.
(C) What makes you sure that miracles cannot occur?
(A) Because they defy the laws of nature.
(C) But if there is a God then couldn't he make an exception to the laws (or as I might say, the patterns) of nature?
(A) But there is no God, and you seem to be assuming there is. Nature is all there is, and that is governed by scientific laws.
Sometimes the argument isn't laid out quite like this, but it's rhetorically shrewd on the part of the atheist because not only are they arguing in a circle but they end up implying that the Christian is the one with the circular reasoning. In fact, the Christian is only starting from the premise that it is possible that God exists and using the evidence of miracles to infer that it's likely that God exists. In contrast, the atheist starts with the assumption that there is no God and uses that to infer that there is no God.
Of course, it's possible that the atheist might argue differently and question the historical evidence and say that it has been fabricated, or corrupted etc., in which case they avoid the circle. But even this can be dangerous because their certainty that it must have been fabricated or corrupted generally flows from their preexisting belief that there is no God. In fact, even if there is only a minuscule chance (for instance) that the entire early church was deceived and/or deceptive, that will be a preferred explanation for the widespread belief in the miraculous events (including the resurrection of Jesus), since the atheist in this discussion is already convinced that miracles cannot occur.
But I will reserve my thoughts on what constitutes sufficient evidence to believe an account of a miracle for another post. At this point I'll just hint at my suspicion that Carl Sagan's saying that "extraordinary claims require extraordinary evidence" might also be straying into circular territory when applied to reject the historical evidence for miracles.