Area of a circle passing through two vertices of a parallelogram touching one edge. For reference:

Let $ABCD$ be a parallelogram, $AB = 6, BC= 10$ and $AC = 14$ ; traces a circle passing through the vertices $C$ and $D$ with $AD$ being tangent and this circle is $BC$ a secant. Calculate the area of ​​the circle. (Answer: $12\pi$)


My progress:
$AD^2 =AI\cdot AC \implies 10^2=AI\cdot14$
$ \implies AI = \dfrac{50}{7} ,\  IC = \dfrac{48}{7}$
In $\triangle ABC$
$14^2=6^2+10^2-2\cdot6\cdot10\cdot\cos \angle B\implies
\cos \angle B =-\dfrac{1}{2}\therefore \angle B =120^o  $
....I can't see many options????

The center $(x,y)$ of the circle has obvious $x=10$; the ordinate $y$ is given by the intersection of lines $x=10$ and the perpendicular to the segment $DC$ at its midpoint (and this ordinate is clearly the radius also by the tangency).
$► \cos(\angle{ABC})=-\dfrac12\Rightarrow \angle{BAD}=60^{\circ}$.
It follows:
$►$ Coordinates $D=(10,0),C=(13,3\sqrt3)$, midpoint of $DC=(11.5,1.5\sqrt3)$
$►$ Line $DC$ has pente $\sqrt3$ then the perpend. has equation $2y-3\sqrt3=-\dfrac{1}{\sqrt3}(2x-23)$ which gives for $x=10$ the value $y=2\sqrt3$.
$►$ Finally the area is $\pi r^2=12\pi$.
