Can a circle be circumscribed around quadrilateral $A B C D$, if $\angle A D C=30^{\circ}, A B=3, B C=4, A C=6$?
Apply the Law of Cosines and the property of a cyclic quadrilateral.

## Solution

By the Law of Cosines in triangle $A B C$, we find that

$$
\cos \angle B=\frac{9+15-36}{2 \cdot 3 \cdot 4}=-\frac{11}{24} \neq-\frac{\sqrt{3}}{2} .
$$

Therefore, $\angle B+\angle D \neq 180^{\circ}$. Hence, it is not possible to circumscribe a circle around quadrilateral $A B C D$.

## Answer

No.