Circles with Same Poles are Parallel

Theorem

Let $S$ be a sphere.

Let $C$ and $D$ be circles on $S$ (either great circles or small circles).

Let $C$ and $D$ both have the same pair of poles.

Then $C$ and $D$ are parallel.

Proof

Let the poles of $C$ and $D$ be $A$ and $B$.

$C$ and $D$ each lie embedded in a plane.

Both of these planes by definition are perpendicular to $AB$.

The result follows from Planes Perpendicular to same Straight Line are Parallel.

$\blacksquare$

Sources

• 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $2$. The spherical triangle.