Proof
Let be bisected at (I.10) and cut unequally at .
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),
meeting at and at . Through draw parallel
to or (I.31), meeting extended at .
The complement equals the complement in the square
(I.43). Add to each the square ; then the rectangle plus
the square equals the rectangle plus the same square.
But together with rectangle -equivalent piece
(which equals since and the lines are parallel)
fills the gnomon , plus the square on , equals the
square on . Thus the rectangle together
with the square on equals the square on .
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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.36Proposition I.36Parallelograms which are on equal bases and in the same parallels are equal to one another.
- 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.
- 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) (4)
- 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…
- II.14Proposition II.14To construct a square equal to a given rectilineal figure.
- III.35Proposition III.35If in a circle two straight lines cut one another, the rectangle contained by the segments of the one 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…
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