Proposition III.15 — Of straight lines in a circle the diameter is greatest, and …

Proof

Let

A D

be a diameter, and

B C

any other chord. From the centre

E

drop

E F ⊥ B C

(I.12). By III.3,

F

is the midpoint of

B C

, so

B F = B C /2

. By I.47 in

△ E B F

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

. Since

E B =

radius

= A D /2

(A D /2)^{2} = (B C /2)^{2} + E F^{2}

, hence

B C^{2} = A D^{2} - 4 \cdot E F^{2} < A D^{2}

when

E F > 0

. So the diameter

A D

is the longest chord. For two non-diameter chords with distances

E F_{1} < E F_{2}

from the centre, the same identity gives the chord through

F_{1}

longer than the one through

F_{2}

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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…
III.4Definition III.4In a circle, straight lines are said to be equally distant from the centre when the perpendiculars drawn to them from…
III.5Definition III.5And that straight line is said to be at a greater distance on which the greater perpendicular falls.

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