2.1. Trapezoid $A B C D$ with base $A D=6$ is inscribed in a circle. The tangent to the circle at point $A$ intersects lines $B D$ and $C D$ at points $M$ and $N$ respectively. Find $A N$, if $A B \perp M D$ and $A M=3$.
Answer: 12. Solution. The condition $A B \perp M D$ means that $\angle A B D=90^{\circ}$, that is, the base $A D$ is the diameter of the circle. Since the trapezoid is inscribed, it is isosceles. $\triangle D N A$ and $\triangle D A C$ are similar as right triangles with a common angle. From the equality of angles $\angle C A D$ and $\angle B D A$, the similarity of $\triangle D A C$ and $\triangle M D A$ follows. Therefore, $\triangle D N A \square \triangle M D A$, that is, $\frac{A N}{A D}=\frac{A D}{A M}=2$, hence $A N=12$.