Celestial Equator is Parallel to Geographical Equator
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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.