$$ \text{Each side of the triangle has a resistance of } R_{\text{side}} = \frac{9\,\Omega}{3} = 3\,\Omega. $$

When measuring the equivalent resistance between any two vertices, there are two paths:

A direct path along one side with resistance:

$$ R_1 = 3\,\Omega. $$

An indirect path passing through the other two sides in series:

$$ R_2 = 3\,\Omega + 3\,\Omega = 6\,\Omega. $$

These two paths are in parallel, so the equivalent resistance is calculated by:

$$ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} = \frac{1}{3\,\Omega} + \frac{1}{6\,\Omega} = \frac{2}{6\,\Omega} + \frac{1}{6\,\Omega} = \frac{3}{6\,\Omega}. $$

Thus,

$$ R_{\text{eq}} = \frac{6\,\Omega}{3} = 2\,\Omega. $$

The equivalent resistance across any two vertices of the equilateral triangle is therefore:

$$ 2\,\Omega. $$