Proof
Let two equal circles have equal central angles and
. In radius-chord-radius triangles and
: (equal radii, equal circles)
and (given); by SAS (I.4) the triangles
are congruent and the chords . Equal chords in equal
circles subtend equal arcs (by superposition). For inscribed
angles, double them via III.20 to reduce to the central-angle case.
Knowledge graph · drag to pan, scroll to zoom, click a node to navigate
Full neighborhood
Depends on (4)
- 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…
- I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight…
- I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…
- III.1Definition III.1Equal circles are those whose diameters are equal, or whose radii are equal.
Required by (dependents) (3)
- III.27Proposition III.27In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or…
- III.28Proposition III.28In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to…
- IV.11Proposition IV.11In a given circle to inscribe an equilateral and equiangular pentagon.
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.
Sign in with ORCID to annotate this claim.