Proposition: 6.33: Angles in Circles have Same Ratio as Arcs

(Proposition 33 from Book 6 of Euclid's “Elements”)

In equal circles, angles have the same ratio as the (ratio of the) circumferences on which they stand, whether they are standing at the centers (of the circles) or at the circumferences. * Let $ABC$ and $DEF$ be equal circles, and let $BGC$ and $EHF$ be angles at their centers, $G$ and $H$ (respectively), and $BAC$ and $EDF$ (angles) at their circumferences. * I say that as circumference $BC$ is to circumference $EF$, so angle $BGC$ (is) to $EHF$, and (angle) $BAC$ to $EDF$.

fig33e

Modern Formulation

This is a generalization of Prop 3.26. In congruent circles, the central angles or angles standing upon arcs are proportional if the arcs have proportional lengths.

Proofs: 1

Proofs: 1 2 3


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References

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016