Equivalence of Definitions of Small Circle

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Theorem

The following definitions of the concept of Small Circle are equivalent:

Definition $1$

A small circle on a sphere $S$ is defined as the intersection of $S$ with a plane which does not pass through the center of $S$.

Definition $2$

A small circle on a sphere $S$ is defined as a circle on the surface of $S$ whose center is not at the center of $S$.


Proof

Let $S$ be a sphere.


Definition $(1)$ implies Definition $(2)$

Let $C$ be a small circle on $S$ by definition $1$.


Recall definition $1$ of Small Circle:

A small circle on a sphere $S$ is defined as the intersection of $S$ with a plane which does not pass through the center of $S$.



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Thus $C$ is a small circle on $S$ by definition $2$.

$\Box$


Definition $(2)$ implies Definition $(1)$

Let $C$ be a small circle on $S$ by definition $2$.


Recall definition $2$ of Small Circle:

A small circle on a sphere $S$ is defined as a circle on the surface of $S$ whose center is not at the center of $S$.



This needs considerable tedious hard slog to complete it.
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When this work has been completed, you may remove this instance of {{Finish}} from the code.
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Thus $C$ is a small circle on $S$ by definition $1$.

$\blacksquare$

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