Proof
Let be bisected at (I.10) and produced to , so that
is the half plus the added segment . Describe on the
square (I.46), join , and through draw parallel
to or (I.31), meeting at and at .
Through draw parallel to or (I.31), and through
draw parallel to or (I.31).
As in II.5, the complement equals the complement (I.43).
Adding the square (which is the square on ) to both
of (the rectangle on , ) shows that the rectangle
together with the square on equals the gnomon plus
the small square, which is the square on . Hence
as required.
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Full neighborhood
Depends on (9)
- I.10Proposition I.10To bisect a given finite straight line.
- I.31Proposition I.31Through a given point to draw a straight line parallel to a given straight line.
- I.34Proposition I.34In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.
- I.43Proposition I.43In any parallelogram the complements of the parallelograms about the diameter are equal to one another.
- I.46Proposition I.46On a given straight line to describe a square.
- II.5Proposition II.5If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole…
- 2Common notion 2If equals be added to equals, the wholes are equal.
- II.1Definition II.1Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.
- II.2Definition II.2And in any parallelogrammic area let any one whatever of the parallelograms about its diameter, with the two…
Required by (dependents) (5)
- II.11Proposition II.11To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the…
- III.36Proposition III.36If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut…
- 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.3Proposition XIII.3If a straight line be cut in extreme and mean ratio, the square on the lesser segment added to the half of the greater…
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