If two triangles standing on equal bases have supplementary vertical angles, their circumcircles are—

Equal bases subtend equal chords, and supplementary vertical angles ($\theta$ and $180^\circ-\theta$) subtend the same chord on circles of identical radius, since the circumradius is $R=\frac{a}{2\sin\theta}$ and $\sin\theta=\sin(180^\circ-\theta)$. Hence the two circumcircles have equal radii and are equal circles.