Proposition VI.33 — In equal circles angles have the same ratio as the circumfer…

Proof

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).

Knowledge graph · drag to pan, scroll to zoom, click a node to navigate

Full neighborhood

Depends on (3)

III.20Proposition III.20In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same…
III.27Proposition III.27In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or…
V.5Definition V.5Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any…

Discussion

No replications, contradictions, or comments registered yet for this claim.

Replicate or annotate this claim

Replicate to register a fresh attempt; contradict, extend, or comment otherwise. Authors can post a claim-retraction with the reason taxonomy from RRP-0020.