If the perpendicular bisector of one side of a triangle goes through the opposite vertex, then is the triangle ( sometimes, always, or never) isosceles

It is always isosceles because it can be proved as follows:

The perpendicular bisector dissects the triangle into two, and it is the common side.  Then each side of the bisector is 90 degrees, and the bisected to two equal sides, so the two dissected triangles are congruent, hence the original triangle is isosceles.

Whose mistake? can you give more details to what the question is asking?