6. It is known that the lengths of the sides of a convex quadrilateral are respectively $a=4, b=5, c=6, d=7$. Find the radius $R$ of the circle circumscribed around this quadrilateral. Provide the integer part of $R^{2}$ as the answer.
Solution. By the cosine theorem, we express the length of the diagonal:

$$
l^{2}=a^{2}+b^{2}-2 a b \cos \gamma, l^{2}=c^{2}+d^{2}-2 c d \cos (\pi-\gamma)
$$

From this, we get $\cos \gamma=\frac{a^{2}+b^{2}-c^{2}-d^{2}}{2(a b+c d)}$. Since $R=\frac{l}{2 \sin \gamma^{\prime}}$, we obtain

$$
R^{2}=\frac{l^{2}}{4\left(1-\cos ^{2} \gamma\right)}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_1dc1b99c67a7c36b0a7dg-3.jpg?height=329&width=320&top_left_y=2005&top_left_x=1553)

For the given side lengths, $R^{2}=\frac{2074799}{131040}$.

Answer: 15.