---
book: 3
number: 16
id: "III.16"
kind: "theorem"
uses: ["[[book-1/proposition-5]]", "[[book-1/proposition-17]]", "[[book-1/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.16"
license: "CC-BY-SA-4.0"
---

# III.16

*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; further the angle of the semicircle is greater*, *and the remaining angle less*, *than any acute rectilineal angle*.

## Proof

Let *ABC* be a circle about *D* as centre and *AB* as diameter; I say that the straight line drawn from *A* at right angles to *AB* from its extremity will fall outside the circle.

For suppose it does not, but, if possible, let it fall within as *CA*, and let *DC* be joined.

Since *DA* is equal to *DC*, the angle *DAC* is also equal to the angle *ACD*. [[book-1/proposition-5|I. 5]]

But the angle *DAC* is right; therefore the angle *ACD* is also right: thus, in the triangle *ACD*, the two angles *DAC*, *ACD* are equal to two right angles: which is impossible. [[book-1/proposition-17|I. 17]]

Therefore the straight line drawn from the point *A* at right angles to *BA* will not fall within the circle.

Similarly we can prove that neither will it fall on the circumference; therefore it will fall outside.

Let it fall as *AE*; I say next that into the space between the straight line *AE* and the circumference *CHA* another straight line cannot be interposed.

For, if possible, let another straight line be so interposed, as *FA*, and let *DG* be drawn from the point *D* perpendicular to *FA*.

Then, since the angle *AGD* is right, and the angle *DAG* is less than a right angle, *AD* is greater than *DG*. [[book-1/proposition-19|I. 19]]

But *DA* is equal to *DH*; therefore *DH* is greater than *DG*, the less than the greater: which is impossible.

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

I say further that the angle of the semicircle contained by the straight line *BA* and the circumference *CHA* is greater than any acute rectilineal angle, and the remaining angle contained by the circumference *CHA* and the straight line *AE* is less than any acute rectilineal angle.

For, if there is any rectilineal angle greater than the angle contained by the straight line *BA* and the circumference *CHA*, and any rectilineal angle less than the angle contained by the circumference *CHA* and the straight line *AE*, then into the space between the circumference and the straight line *AE* a straight line will be interposed such as will make an angle contained by straight lines which is greater than the angle contained by the straight line *BA* and the circumference *CHA*, and another angle contained by straight lines which is less than the angle contained by the circumference *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 *BA* and the circumference *CHA*, nor yet any acute angle contained by straight lines which is less than the angle contained by the circumference *CHA* and the straight line *AE*.—

Porism. 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. Q. E. D.