Proposition III.28 — In equal circles equal straight lines cut off equal circumfe…

Proof

Let

A B

and

C D

be equal chords in equal circles with centres

E

F

. In

△ A B E

and

△ C D F

A B = C D

A E = B E = C F = D F

(radii, equal circles). By SSS (I.8) the triangles are congruent and

∠ A E B = ∠ C F D

. By III.26 the arcs are equal. The corresponding major arcs (complements in the equal circles) are also equal.

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Full neighborhood

Depends on (3)

III.26Proposition III.26In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences.
I.8Proposition I.8If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they…
III.1Definition III.1Equal circles are those whose diameters are equal, or whose radii are equal.

Required by (dependents) (1)

III.30Proposition III.30To bisect a given arc.

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