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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 equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less.

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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· 58 files

In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less.

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.