First, why this didn't work.Originally Posted by e-mom
1) You proved that the wrong pair of sides were equal. (We need AB=AC, you had AB=BC).
2) The reason this failed was the assumption that the perpendicular bisector of AC intersects the point B. This is not true in general. (Draw a more lopsided triangle and it should be clear.)
I would suggest that you bisected the wrong thing. I would start over with your triangle, bisect the angle A, and extend the bisector to intersect the line BC at point D. (Note: We canNOT assume BD=CD here. We would need to prove that.) The goal would then to be to prove that triangle ADB is congruent with triangle ADC. (Hint: Look for an angle-side-angle proof.)