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.