Constructing a circle through 2 points

Question

We have a triangle ABC with a circumscribed circle. Somewhere between BC we place a point D. There is a circle which goes through D and whose tangent at AB is A. This circle also intersects the circumscribed circle of ABC at a point E. Construct it.

So we're just going to construct a circle through points A and D, then see where it intersects ABC. So I originally thought that every point on the bisector of AD would work, but apparently not.

The answer sheet says that the center of the circle is the intersection of the bisector of AD and the line perpendicular to AB through the point A.

Why the perp line through A? I don't understand..

Answer 1

It doesn't look like you have enough information to determine $E$ from what you have written. Since any three non-collinear points determine a circle, you could select $E$ anywhere on the circumscribed circle (except for $A$ and the other end of the diameter through $A$ and $D$) and then there'd be a circle through $A$, $D$ and $E$.

The answer sheet says that it is the intersection of the bisector of AD and the line perpendicular to AB through the point A.

That point generally won't like on the circumscribed circle at all. It is, however, the center of the circle that goes through $D$ and whose tangent at $A$ is $AB$.