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

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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}

.