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

; join

. Since

is a diameter and

on the circle, by III.31 (proved independently below)

is right. But

is also right by construction; the sum of two angles of

is then two right angles, leaving no positive angle at

— contradicting I.17. Hence

meets the circle only at

. The "no other line interposable" follows from the uniqueness of the perpendicular (I.11): any line through

not perpendicular to

makes a non-right angle and cuts the circle.