Locus of the point of contact of tangent.

Question

Let $A, B, C$ be three points on a straight line, $B$ lying between $A$ and $C$. Consider all circles passing through $B$ and $C$. The points of contact of the tangents from $A$ to these circles lie on?

The answer given is circle but I'm unable to prove it. Can anyone help?

Answer 1

By power of a point, the lengths of any tangents from A to the circles passing through BC are always constant [$= \sqrt {AB.AC}$, (modified) ]. Hence the required locus is on the circle centered at A with radius $= \sqrt {AB.AC}$.

Answer 2

Let $K$ be the touching point.

Thus, $$AK^2=AB\cdot AC,$$ which says that the locus placed on the circle with a radius $\sqrt{AB\cdot AC}$ and with the center $A$.