In the same / equal circle: central angle / arc / chord are pairwise equivalent

Dependencies: Protractor axiom, SAS congruence.

Statement

Let $\odot O$ be a given circle of radius $R$, with $A$, $B$, $C$, $D$ on $\odot O$. Consider the following three statements:

1. Equal central angles: $\angle AOB = \angle COD$;
2. Equal arcs: $\stackrel{\frown}{AB} = \stackrel{\frown}{CD}$ (subtended in the same orientation);
3. Equal chords: $|AB| = |CD|$.

In a single circle (or in two equal circles), these three statements are pairwise equivalent: any one implies the other two. Formally: