Call the point where the vertical line intersects BC point O. Assuming the vertical line is perpendicular to BC (is it?), then angle BOA = COA = 90 degrees. Then you have triangles ABO and ACO are equivalent from side-angle-side. Hence AB = AC.
Sorry, there was a right angle. I forgot to draw it.
Well then, next time please review before posting making sure all relevant information is given.
The only triangle in which an altitude is a bisector of its base is an isosceles triangle.