Proof
Let be the given straight line. Describe on the square
(I.46). Bisect at (I.10) and join . Produce
to in the direction of (Postulate 2), and lay off
on produced so that (I.3, taking as
the standard length). On describe the square (I.46);
produce to meet at .
Then by II.6 applied to bisected at with extension ,
the rectangle on , together with the square on equals
the square on . But , so this rectangle plus square
on equals the square on , which by I.47 (in
, right-angled at ) equals the square on
plus the square on . Subtracting the square on from both
sides (Common Notion 3):
\[
CF \cdot FA \;=\; AB^2.
\]
The rectangle on , (= since )
equals the square on . Subtracting the common rectangle on
, from both, the square on equals the rectangle
on and . Setting on (point
on with ) gives the desired section: .
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Full neighborhood
Depends on (10)
- I.3Proposition I.3Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
- I.10Proposition I.10To bisect a given finite straight line.
- I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
- I.46Proposition I.46On a given straight line to describe a square.
- I.47Proposition I.47In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides…
- II.6Proposition II.6If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the…
- 1Common notion 1Things which are equal to the same thing are also equal to one another.
- 2Common notion 2If equals be added to equals, the wholes are equal.
- 3Common notion 3If equals be subtracted from equals, the remainders are equal.
- 2Postulate 2To produce a finite straight line continuously in a straight line.
Required by (dependents) (7)
- IV.10Proposition IV.10To construct an isosceles triangle having each of the angles at the base double of the remaining one.
- VI.30Proposition VI.30To cut a given finite straight line in extreme and mean ratio.
- XIII.1Proposition XIII.1If a straight line be cut in extreme and mean ratio, the square on the greater segment added to the half of the whole…
- XIII.2Proposition XIII.2If the square on a straight line be five times the square on a segment of it, then, when the double of the said segment…
- XIII.4Proposition XIII.4If a straight line be cut in extreme and mean ratio, the square on the whole and the square on the lesser segment…
- XIII.5Proposition XIII.5If a straight line be cut in extreme and mean ratio, and there be added to it a straight line equal to the greater…
- XIII.6Proposition XIII.6If a rational straight line be cut in extreme and mean ratio, each of the segments is the irrational straight line…
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