1. Connect the common points of the circle $x^{2}+(y-1)^{2}=1$ and the ellipse $9 x^{2}+(y+1)^{2}=9$ with line segments. The resulting figure is
A. Line segment
B. Scalene triangle
C. Equilateral triangle
D. Quadrilateral
Solving
\[
\left\{
\begin{array}{l}
x^{2}+(y-1)^{2}=1, \\
x^{2}+\frac{(y+1)^{2}}{9}=1
\end{array}
\Rightarrow(y+1)^{2}=9(y-1)^{2} \Rightarrow y=2 \text{ or } y=\frac{1}{2}.
\right.
\]
Thus, the three intersection points are $(0,2),\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right),\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, which form an equilateral triangle when connected. Therefore, the answer is $C$.