Proof
Let the straight line be cut at random at . Describe on
the square (I.46), and draw the diagonal . Through
draw parallel to either or (I.31), meeting at
and at . Through draw parallel to either
or (I.31), meeting at and at .
Since is parallel to and falls on them, the exterior
angle equals the interior and opposite
(I.29). But since (I.5 applied
to the isoceles right triangle inside the square). Hence
, so (I.6), and therefore
is equilateral. Since it has a right angle at , it is a square
on (Definition I.22). By the same reasoning is the
square on .
The complements and in the square are equal
rectangles by I.43; each is contained by and (since
, , etc.), so each is the rectangle on , .
The four pieces sum to the whole (Common Notion 2):
\[
AB^2 \;=\; AC^2 + CB^2 + 2\cdot(AC\cdot CB),
\]
which is in geometric form.
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Full neighborhood
Depends on (11)
- I.5Proposition I.5In isosceles triangles the angles at the base are equal to one another; and if the equal straight lines be produced…
- I.6Proposition I.6If in a triangle two angles are equal to one another, the sides which subtend the equal angles will also be equal to…
- I.29Proposition I.29A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle…
- 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.
- 2Common notion 2If equals be added to equals, the wholes are equal.
- I.22Definition I.22Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is…
- 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.7Proposition II.7If a straight line be cut at random, the square on the whole and that on one of the segments both together are equal to…
- II.8Proposition II.8If a straight line be cut at random, four times the rectangle contained by the whole and one of the segments together…
- II.12Proposition II.12In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides…
- 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…
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