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

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In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another.

Let

A B

and

C D

be chords with

A B = C D

. From the centre

E

drop perpendiculars

E F

to

A B

and

E G

to

C D

(I.12). By III.3,

A F = F B = A B /2

and

C G = G D = C D /2

, so

A F = C G

. Join

E A

,

E C

(both radii, so equal). In right triangles

△ E A F

and

△ E C G

(right angles at

F

,

G

), I.47 gives

E A^{2} = A F^{2} + E F^{2}

and

E C^{2} = C G^{2} + E G^{2}

. Subtracting (Common Notion 3) and using

E A = E C

,

A F = C G

gives

E F^{2} = E G^{2}

, hence

E F = E G

. Conversely, if

E F = E G

, the same I.47 identity gives

A F = C G

and hence

A B = C D

.