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

. From

draw lines

,

to the circumference. Join

,

. In

:

(I.20). But

(radii) and

, so

; hence the line

along the diameter towards the centre is longer than any other. The line

on the other side is similarly the shortest. For intermediate lines

vs

with

closer to

than

, the SAS inequality I.24 in the radius-line-radius triangles gives

when

.