Circles with Same Poles are Parallel
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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.