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Proof: By Euclid

(related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)

• For (if) not then, if possible, let it fall inside, like $CA$ (in the figure), and let $DC$ have been joined. Since $DA$ is equal to $DC$, angle $DAC$ is also equal to angle $ACD$ [Prop. 1.5].
• And $DAC$ (is) a right angle.
• Thus, $ACD$ (is) also a right angle.
• So, in triangle $ACD$, the two angles $DAC$ and $ACD$ are equal to two right angles.
• The very thing is impossible [Prop. 1.17].
• Thus, the (straight line) drawn from point $A$, at right angles to $BA$, will not fall inside the circle.
• So, similarly, we can show that neither (will it fall) on the circumference.
• Thus, (it will fall) outside (the circle).
• Let it fall like $AE$ (in the figure).
• So, I say that another straight line cannot be inserted into the space between the straight line $AE$ and the circumference $CHA$.
• For, if possible, let it be inserted like $FA$ (in the figure), and let $DG$ have been drawn from point $D$, perpendicular to $FA$ [Prop. 1.12].
• And since $AGD$ is a right angle, and $DAG$ (is) less than a right angle, $AD$ (is) thus greater than $DG$ [Prop. 1.19].
• And $DA$ (is) equal to $DH$.
• Thus, $DH$ (is) greater than $DG$, the lesser than the greater.
• The very thing is impossible.
• Thus, another straight line cannot be inserted into the space between the straight line ($AE$) and the circumference.
• And I also say that the semi-circular angle contained by the straight line $BA$ and the circumference $CHA$ is greater than any acute rectilinear angle whatsoever, and the remaining (angle) contained by the circumference $CHA$ and the straight line $AE$ is less than any acute rectilinear angle whatsoever.
• For if any rectilinear angle is greater than the (angle) contained by the straight line $BA$ and the circumference $CHA$, or less than the (angle) contained by the circumference $CHA$ and the straight line $AE$, then a straight line can be inserted into the space between the circumference $CHA$ and the straight line $AE$ - anything which will make (an angle) contained by straight lines greater than the angle contained by the straight line $BA$ and the circumference $CHA$, or less than the (angle) contained by the circumference $CHA$ and the straight line $AE$.
• But (such a straight line) cannot be inserted.
• Thus, an acute (angle) contained by straight lines cannot be greater than the angle contained by the straight line $BA$ and the circumference $CHA$, neither (can it be) less than the (angle) contained by the circumference $CHA$ and the straight line $AE$.
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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"