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Source — Euclid's Elements, encoded as an rrxiv paper — rrxiv

Source — Euclid's Elements, encoded as an rrxiv paper

In equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centres or at the circumferences.

Use III.27 (equal arcs subtend equal central angles in equal circles) to set up an equimultiples test: for any positive integers

m

n

m

copies of one angle correspond to

m

copies of its arc, and the order of

m α

versus

n β

matches the order of

m \cdot arc (α)

versus

n \cdot arc (β)

. This is exactly Definition V.5 for

α : β = arc (α) : arc (β)

. The inscribed-angle case follows from III.20 (inscribed angle is half the central).

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In equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centres or at the circumferences.

Use III.27 (equal arcs subtend equal central angles in equal circles) to set up an equimultiples test: for any positive integers

m

n

m

copies of one angle correspond to

m

copies of its arc, and the order of

m α

versus

n β

matches the order of

m \cdot arc (α)

versus

n \cdot arc (β)

. This is exactly Definition V.5 for

α : β = arc (α) : arc (β)

. The inscribed-angle case follows from III.20 (inscribed angle is half the central).