Symmetry means, hmmm usually a mirror image.
So when something is symmetry to another, is a way to say, one is the mirror image of the other, or namely they're just identical twins.
Now, if know that B is the midpoint, by definition is cutting AC into two equal halves, so AB is the mirror image of BC, so they're identical twins, there are no flaws, Q.E.D.
Check the picture below.