Please help me with this question! I can't seem to get full marks! I'll give out brainiest! A, B and C are points on the circumference of a circle. XY is a tangent to the circle at the point A.

Angle BAY=72 and angle ABC=54.

Prove that triangle ABC is an isosceles triangle.

You must give a reason for any statement or any calculation you carry out.

This is my working out:
Angle BCA = 72 degrees (Alternate segment theorem)
A tangent meets a radius at 90 degrees.
Angle BCA + angle ABC = 72 + 54= 126 degrees (Angles in a triangle add up to 180 degrees)
Angle BAC = 180 - (72 + 54) = 54 degrees (angles in a triangle add up to 180 degrees)
Since angle BAC = angle BCA so the triangle is isosceles. AC= BC hence BAC = BCA because they are congruent angles which makes triangle ABC an isosceles triangle.
AC= BC because both of these are equal to the radius of the circle.
Angles in a semicircle are 90 degrees.