Points A and C lie on a circle. The tangents to the circle drawn through these points intersect at point B, AC = AB.

Points A and C lie on a circle. The tangents to the circle drawn through these points intersect at point B, AC = AB. Prove that the bisector of angle ACB passes through the midpoint of segment AB.

The tangents intersect at point B, which means that the segments of the tangents BA and BC are equal.
From the data of the task it follows that VA is also equal to the AC.
Thus, all three sides of the triangle ACB are equal to each other, AC = AB = BC.
It follows from this that the ACB triangle is equilateral.
Since in an equilateral triangle the bisector, median and height are equal and divide the angles and opposite sides in half, the bisector of angle ACB will pass through the middle of the segment AB.
Q.E.D.