1. Given is $\triangle A B C$ for which $A B=A C=18$ and $B C=4$. A circle $k$ with radius 7 is observed, passing through points $B$ and $C$, and its center, point $O$, is located inside $\triangle A B C$. Tangents from point $A$ to $k$ touch it at points $N$ and $M$. Find the area of quadrilateral OMAN.
1. Let $K$ be the midpoint of side $BC$. Since $\triangle ABC$ is isosceles, $AK$ is the perpendicular bisector of side $BC$, and point $O$ lies on it. Now we have $OK = \sqrt{OC^2 - CK^2} = \sqrt{49 - 4} = 3\sqrt{5}$ and $AK = \sqrt{AC^2 - CK^2} = \sqrt{324 - 4} = 8\sqrt{5}$, and thus $AO = AK - OK = 5\sqrt{5}$. Further, since the angle at vertex $M$ in $\triangle OMA$ is right, we calculate $AM = \sqrt{AO^2 - OM^2} = \sqrt{125 - 49} = 2\sqrt{19}$. Finally, we have $P(OMAN) = 2P(\triangle OMA) = 2 \cdot \frac{OM \cdot AM}{2} = 7 \cdot 2\sqrt{19} = 14\sqrt{19}$.