1. In $\triangle A B C$, let the lengths of the sides opposite to the interior angles $A, B, C$ be $a, b, c$ respectively. Proposition $p: B+C=2 A$, and $b+c=2 a$; Proposition $q: \triangle A B C$ is an equilateral triangle. Then proposition $p$ is a
A. Sufficient and necessary condition
B. Sufficient condition but not a necessary condition
C. Necessary condition but not a sufficient condition
D. Neither a sufficient condition nor a necessary condition
$$
\sin B+\sin \left(120^{\circ}-B\right)=\frac{3}{2} \sin B+\frac{\sqrt{3}}{2} \cos B=\sqrt{3} \sin \left(B+30^{\circ}\right)=2 \sin 60^{\circ}=\sqrt{3} \Rightarrow B=60^{\circ}
$$
$\Rightarrow \triangle A B C$ is an equilateral triangle. So the answer is $A$.