Euclid's Proposition III.14 proves that chords are of equal length iff they are equidistant from the center. Euclid's proof is long and complicated, repeatedly invoking the Pythagorean Theorem, which ultimately depends on the Parallel Postulate.
This is very surprising, given that a very simple proof not dependent on the Parallel Postulate is available (and Euclid always avoids the Parallel Postulate where possible, e.g. the first 30 propositions of Book I):
Let $AB$ and $A'B'$ be equal length chords in $\bigcirc O$. Then $OA \cong OA', OB \cong OB', AB \cong A'B'$, so $\triangle ABO \cong \triangle A'B'O$ by SSS, and their altitudes are equal. QED.
One might object that we haven't explicitly proven that congruent triangles have equal altitudes. But this too is easily proven, esp. since Euclid has already proven that this altitude bisects the chord (III.3):
From $O$, drop the altitude to $AB$ at $M$ and to $A'B'$ at $M'$. Since $AB \cong A'B'$, $MA$ (which is half of $AB$) is congruent to $MA'$ (half of $A'B'$), so $\triangle MAO \cong \triangle M'A'O$ by ASA, and $MO \cong M'O$.
Finally, the converse comes almost as easy:
Given $OM \cong OM'$, then $\triangle OMA \cong OM'A'$ and $\triangle OMB \cong OM'B'$ by HL (aka RASS), so $AB \cong A'B'$.
Although Euclid has not yet proven RASS, the proof is easy and does not require the Parallel Postulate.
Questions:
- Are the proofs above correct? Aside from why Euclid didn't use them, can you verify or critique these proofs?
- Why didn't Euclid use them, given that they are simpler, shorter, more explicit, and don't require the Parallel Postulate?