---
book: 3
number: 17
id: "III.17"
kind: "construction"
uses: ["[[book-3/proposition-1]]", "[[book-1/proposition-4]]", "[[book-3/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:3.prop.17"
license: "CC-BY-SA-4.0"
---

# III.17

*From a given point to draw a straight line touching a given circle*.

## Proof

Let *A* be the given point, and *BCD* the given circle; thus it is required to draw from the point *A* a straight line touching the circle *BCD*.

For let the centre *E* of the circle be taken; [[book-3/proposition-1|III. 1]] let *AE* be joined, and with centre *E* and distance *EA* let the circle *AFG* be described; from *D* let *DF* be drawn at right angles to *EA*, and let *EF*, *AB* be joined; I say that *AB* has been drawn from the point *A* touching the circle *BCD*.

For, since *E* is the centre of the circles *BCD*, *AFG*, *EA* is equal to *EF*, and *ED* to *EB*; therefore the two sides *AE*, *EB* are equal to the two sides *FE*, *ED*: and they contain a common angle, the angle at *E*; therefore the base *DF* is equal to the base *AB*, and the triangle *DEF* is equal to the triangle *BEA*, and the remaining angles to the remaining angles; [[book-1/proposition-4|I. 4]] therefore the angle *EDF* is equal to the angle *EBA*.

But the angle *EDF* is right; therefore the angle *EBA* is also right.

Now *EB* is a radius; and the straight line drawn at right angles to the diameter of a circle, from its extremity, touches the circle; [[book-3/proposition-16|III. 16, Por.]] therefore *AB* touches the circle *BCD*.

Therefore from the given point *A* the straight line *AB* has been drawn touching the circle *BCD*.