triangle: distance between segment midpoint and opposite vertex

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Assuming I have 3 vertices A, B, C forming a triangle as shown in the picture and the locations (e.g. in 2D) of A and B as well as the distances AC and BC (and obviously AB) are known (but not the location of C). Is there an elegant way of calculating the distance MC where M is the segment midpoint between A and B? My ugly solution would be to first find the position of C and then simply calculate the distance between C and M. However, I was wondering if there is a more elegant way given AB, AC and BC.

triangle