Line at Right Angles to Diameter of Circle

Theorem

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

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

The angle of the semicircle is greater, and the remaining angle less than, any acute rectilineal angle.

Porism

The straight line drawn at right angles to the diameter of a circle from its extremity touches the circle.

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 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 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 is Proposition 16 of Book III of Euclid's The Elements.