---
book: 3
number: 33
id: "III.33"
kind: "construction"
uses: ["[[book-1/proposition-4]]", "[[book-3/proposition-16]]", "[[book-3/proposition-32]]", "[[book-3/proposition-31]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.33"
license: "CC-BY-SA-4.0"
---

# III.33

*On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle*.

## Proof

Let *AB* be the given straight line, and the angle at *C* the given rectilineal angle; thus it is required to describe on the given straight line *AB* a segment of a circle admitting an angle equal to the angle at *C*.

The angle at *C* is then acute, or right, or obtuse.

First let it be acute, and, as in the first figure, on the straight line *AB*, and at the point *A*, let the angle *BAD* be constructed equal to the angle at *C*; therefore the angle *BAD* is also acute.

Let *AE* be drawn at right angles to *DA*, let *AB* be bisected at *F*, let *FG* be drawn from the point *F* at right angles to *AB*, and let *GB* be joined.

Then, since *AF* is equal to *FB*, and *FG* is common, the two sides *AF*, *FG* are equal to the two sides *BF*, *FG*; and the angle *AFG* is equal to the angle *BFG*; therefore the base *AG* is equal to the base *BG*. [[book-1/proposition-4|I. 4]]

Therefore the circle described with centre *G* and distance *GA* will pass through *B* also.

Let it be drawn, and let it be *ABE*; let *EB* be joined.

Now, since *AD* is drawn from *A*, the extremity of the diameter *AE*, at right angles to *AE*, therefore *AD* touches the circle *ABE*. [[book-3/proposition-16|III. 16, Por.]]

Since then a straight line *AD* touches the circle *ABE*, and from the point of contact at *A* a straight line *AB* is drawn across in the circle *ABE*, the angle *DAB* is equal to the angle *AEB* in the alternate segment of the circle. [[book-3/proposition-32|III. 32]]

But the angle *DAB* is equal to the angle at *C*; therefore the angle at *C* is also equal to the angle *AEB*.

Therefore on the given straight line *AB* the segment *AEB* of a circle has been described admitting the angle *AEB* equal to the given angle, the angle at *C*.

Next let the angle at *C* be right; and let it be again required to describe on *AB* a segment of a circle admitting an angle equal to the right angle at *C*.

Let the angle *BAD* be constructed equal to the right angle at *C*, as is the case in the second figure; let *AB* be bisected at *F*, and with centre *F* and distance either *FA* or *FB* let the circle *AEB* be described.

Therefore the straight line *AD* touches the circle *ABE*, because the angle at *A* is right. [[book-3/proposition-16|III. 16, Por.]]

And the angle *BAD* is equal to the angle in the segment *AEB*, for the latter too is itself a right angle, being an angle in a semicircle. [[book-3/proposition-31|III. 31]]

But the angle *BAD* is also equal to the angle at *C*.

Therefore the angle *AEB* is also equal to the angle at *C*.

Therefore again the segment *AEB* of a circle has been described on *AB* admitting an angle equal to the angle at *C*.

Next, let the angle at *C* be obtuse; and on the straight line *AB*, and at the point *A*, let the angle *BAD* be constructed equal to it, as is the case in the third figure; let *AE* be drawn at right angles to *AD*, let *AB* be again bisected at *F*, let *FG* be drawn at right angles to *AB*, and let *GB* be joined.

Then, since *AF* is again equal to *FB*, and *FG* is common, the two sides *AF*, *FG* are equal to the two sides *BF*, *FG*; and the angle *AFG* is equal to the angle *BFG*; therefore the base *AG* is equal to the base *BG*. [[book-1/proposition-4|I. 4]]

Therefore the circle described with centre *G* and distance *GA* will pass through *B* also; let it so pass, as *AEB*.

Now, since *AD* is drawn at right angles to the diameter *AE* from its extremity, *AD* touches the circle *AEB*. [[book-3/proposition-16|III. 16, Por.]]

And *AB* has been drawn across from the point of contact at *A*; therefore the angle *BAD* is equal to the angle constructed in the alternate segment *AHB* of the circle. [[book-3/proposition-32|III. 32]]

But the angle *BAD* is equal to the angle at *C*.

Therefore the angle in the segment *AHB* is also equal to the angle at *C*.

Therefore on the given straight line *AB* the segment *AHB* of a circle has been described admitting an angle equal to the angle at *C*. Q. E. F.