Proposition III.14 — In a circle equal straight lines are equally distant from th…

Proof

Let

A B

and

C D

be chords with

A B = C D

. From the centre

E

drop perpendiculars

E F

A B

and

E G

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

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Depends on (5)

III.3Proposition III.3If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at…
I.12Proposition I.12To draw a perpendicular straight line to a given infinite straight line from a given point not on it.
I.47Proposition I.47In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides…
3Common notion 3If equals be subtracted from equals, the remainders are equal.
III.4Definition III.4In a circle, straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from…

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