Theorem: (Equator Projection)

The image of the equator under the stereographic projection is the

circle center at the South Pole whose radius is twice that of the equator.

Proof:

Given a circle C₁centerd at the point O and radius r whose diameter

NS is perpendicular to tangent line j at point S.

Let E' be the projection of E, where E is any point on the equator, onto j.

We need to show that SE' = 2OE.

The point E lies on the equator, hence 0E perpendicular to NS.

Therefore OE is parallel to j.

Since O is the center of NS and OE is parallel to j then E

is the midpoint of line segment NE' by the Midpoint Theorem.

Also by the midpoint Theorem,

SE' = 2OE.

Q.E.D.