Line at Right Angles to Diameter of Circle

Theorem

In the words of Euclid:

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 of the circle another straight line cannot be interposed; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle.

(The Elements: Book $\text{III}$: Proposition $16$)

Porism

In the words of Euclid:

From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its extremity touches the circle.

(The Elements: Book $\text{III}$: Proposition $16$ : Porism)

Proof

Let $ABC$ be a circle about $D$ as center and $AB$ as diameter.

Suppose that a line from $A$ at right angles to the diameter falls within the circle, e.g. at $AC$.

Let $DC$ be joined.

Since $DA = DC$ we have that $\angle DAC = \angle ACD$.

But by hypothesis the angle $DAC$ is a right angle.

Therefore $\angle ACD$ is also a right angle.

So in $\triangle ACD$, the two angles $\angle DAC$ and $\angle ACD$ equal two right angles.

From Two Angles of Triangle are Less than Two Right Angles this is impossible.

Therefore the straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle.

$\Box$

Let this line be as $AE$.

Suppose there were another straight line $AF$ interposed between the circumference of the circle and $AE$.

Let $DG$ be drawn perpendicular to $AF$ from $D$.

Since $\angle AGD$ is a right angle, the angle $\angle DAG$ is less than a right angle by Two Angles of Triangle are Less than Two Right Angles.

So $AD > DG$ from Greater Angle of Triangle Subtended by Greater Side.

But $AD = DH$ and so $DH > DG$, which is impossible.

Therefore another straight line cannot be interposed into the space between the straight line and the circumference of the circle.

$\Box$

Suppose there were a rectilineal angle greater than:

that contained by the straight line $AB$ and the arc $CHA$ of the circle

and:

any rectilineal angle less than the angle contained by the arc $CHA$ and the straight line $AE$.

Then into the space between the circumference and the straight line $AE$ another straight line can be interposed such as will make an angle contained by straight lines greater than the angle contained by the straight line $AB$ and the arc $CHA$.

Also, another angle contained by the straight lines $AB$ and $AE$ which is less than the angle contained by the arc $CHA$ and the straight line $AE$.

But such a straight line cannot be interposed.

Therefore there will not be any acute angle contained by straight lines which is greater than the angle contained by the straight line $AB$ and the arc $CHA$.

Nor will there be any any acute angle contained by straight lines which is less than the angle contained by the arc $CHA$ and the straight line $AE$.

$\blacksquare$

Historical Note

This proof is Proposition $16$ of Book $\text{III}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{III}$. Propositions