The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed.
Proof
Let AB be a diameter and AC drawn at right angles to AB at
A (I.11). Suppose AC meets the circle at another point D=A; join BD. Since AB is a diameter and D on the
circle, by III.31 (proved independently below) ∠ADB is
right. But ∠DAB is also right by construction; the sum of
two angles of △ABD is then two right angles, leaving no
positive angle at B — contradicting I.17. Hence AC meets the
circle only at A. The "no other line interposable" follows from
the uniqueness of the perpendicular (I.11): any line through A
not perpendicular to AB makes a non-right angle and cuts the
circle.