Celestial Equator is Parallel to Geographical Equator

Theorem

Consider the celestial sphere with observer $O$.

The plane of the celestial equator is parallel to the plane of the geographical equator.

Proof

Let $P$ denote the north celestial pole.

The celestial equator is by definition perpendicular to $OP$.

Also by definition, $OP$ is parallel to Earth's axis.

Also by definition, Earth's axis is perpendicular to the geographical equator.

Hence $OP$ is also perpendicular to the geographical equator.

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 $\text {II}$. The Celestial Sphere: $19$. Declination and hour angle.