[Theorem of the length of a tangent and a secant; the product of the entire secant and its external part [Isosceles, inscribed, and circumscribed trapezoids

A circle passing through the vertices $A, B$, and $C$ of parallelogram $A B C D$ touches the line $A D$ and intersects the line $C D$ at points $C$ and $M$. Find the length of the segment $A D$, if $B M=9$ and $D M=8$.

#
The diagonals of the isosceles trapezoid AMCВ are equal. Triangle BAC is isosceles. Then apply the tangent-secant theorem.

## Answer

$6 \sqrt{2}$.