---
book: 3
number: 8
id: "III.8"
kind: "theorem"
uses: ["[[book-3/proposition-1]]", "[[book-1/proposition-20]]", "[[book-1/proposition-24]]", "[[book-1/proposition-21]]", "[[book-1/proposition-4]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.8"
license: "CC-BY-SA-4.0"
---

# III.8

*If a point be taken outside a circle and from the point straight lines be drawn through to the circle*, *one of which is through the centre and the others are drawn at random*, *then*, *of the straight lines which fall on the concave circumference*, *that through the centre is greatest*, *while of the rest* *the nearer to that through the centre is always greater than the more remote*, *but*, *of the straight lines falling on the convex circumference*, *that between the point and the diameter is least*, *while of the rest the nearer to the least is always less than the more remote*, *and only two equal straight lines will fall on the circle from the point*, *one on each side of the least*.

## Proof

Let *ABC* be a circle, and let a point *D* be taken outside *ABC*; let there be drawn through from it straight lines *DA*, *DE*, *DF*, *DC*, and let *DA* be through the centre; I say that, of the straight lines falling on the concave circumference *AEFC*, the straight line *DA* through the centre is greatest, while *DE* is greater than *DF* and *DF* than *DC*.; but, of the straight lines falling on the convex circumference *HLKG*, the straight line *DG* between the point and the diameter *AG* is least; and the nearer to the least *DG* is always less than the more remote, namely *DK* than *DL*, and *DL* than *DH*.

For let the centre of the circle *ABC* be taken [[book-3/proposition-1|III. 1]], and let it be *M*; let *ME*, *MF*, *MC*, *MK*, *ML*, *MH* be joined.

Then, since *AM* is equal to *EM*, let *MD* be added to each; therefore *AD* is equal to *EM*, *MD*.

But *EM*, *MD* are greater than *ED*; [[book-1/proposition-20|I. 20]] therefore *AD* is also greater than *ED*.

Again, since *ME* is equal to *MF*, and *MD* is common, therefore *EM*, *MD* are equal to *FM*, *MD*; and the angle *EMD* is greater than the angle *FMD*; therefore the base *ED* is greater than the base *FD*. [[book-1/proposition-24|I. 24]]

Similarly we can prove that *FD* is greater than *CD*; therefore *DA* is greatest, while *DE* is greater than *DF*, and *DF* than *DC*.

Next, since *MK*, *KD* are greater than *MD*, [[book-1/proposition-20|I. 20]] and *MG* is equal to *MK*, therefore the remainder *KD* is greater than the remainder *GD*, so that *GD* is less than *KD*.

And, since on *MD*, one of the sides of the triangle *MLD*, two straight lines *MK*, *KD* were constructed meeting within the triangle, therefore *MK*, *KD* are less than *ML*, *LD*; [[book-1/proposition-21|I. 21]] and *MK* is equal to *ML*; therefore the remainder *DK* is less than the remainder *DL*.

Similarly we can prove that *DL* is also less than *DH*; therefore *DG* is least, while *DK* is less than *DL*, and *DL* than *DH*.

I say also that only two equal straight lines will fall from the point *D* on the circle, one on each side of the least *DG*.

On the straight line *MD*, and at the point *M* on it, let the angle *DMB* be constructed equal to the angle *KMD*, and let *DB* be joined.

Then, since *MK* is equal to *MB*, and *MD* is common, the two sides *KM*, *MD* are equal to the two sides *BM*, *MD* respectively; and the angle *KMD* is equal to the angle *BMD*; therefore the base *DK* is equal to the base *DB*. [[book-1/proposition-4|I. 4]]

I say that no other straight line equal to the straight line *DK* will fall on the circle from the point *D*.

For, if possible, let a straight line so fall, and let it be *DN*. Then, since *DK* is equal to *DN*,

while *DK* is equal to *DB*, *DB* is also equal to *DN*, that is, the nearer to the least *DG* equal to the more remote: which was proved impossible.

Therefore no more than two equal straight lines will fall on the circle *ABC* from the point *D*, one on each side of *DG* the least.

Therefore etc.