 Ordinary vertex in a nonconcurrent great-circle arrangement

A finite set of distinct great circles not all passing through a common antipodal pair has a vertex lying on exactly two circles. Its cells are spherical polygons with at least three edges: a digon would have antipodal endpoints, and every circle's defining hemisphere would then have to contain both endpoints, forcing a common axis. With $V-E+F=2$ and $3F\le 2E$, the graph has a vertex of valence less than six. Valences are even and at least four, so one is exactly four. Central projection identifies this with an ordinary intersection in a real projective line arrangement. If affine line intersections are counted only at finite points, parallel-line configurations require a separate qualification.