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P l a n e G e o m e t r y
An Adventure in Language and Logic
based on
CONSTRUCTION OF A SQUARE
Book I. Proposition 46
WE ARE JUST ABOUT READY to prove the Pythagorean theorem, which is about the squares that are drawn on the sides of a right-angled triangle. The following proposition will show that the figure we construct satisfies the definition of a square, and therefore that the figure we have called a "square" actually exists. (See the Commentary on the Definitions.)
On a given straight line to draw a square. |
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Let AB be the given straight line;
we are required to draw a square on AB. |
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From the point A draw AC at right angles to the straight line AB, |
(I. 11) |
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and make AD equal to AB; |
(I. 3) |
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through the point D draw DE parallel to AB; |
(I. 31) |
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and through the point B draw BE parallel to AD. |
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Then ADEB is a square. |
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For, ADEB is by construction a parallelogram; |
(Definition 15) |
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therefore AB is equal to DE, and AD is equal to BE. |
(I. 34) |
But AB is equal to AD. |
(Construction) |
Therefore the four straight lines AD, DE, EB, BA are equal to one another, so that the parallelogram ADEB is equilateral. |
(Axiom 1) |
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Next, all its angles are right angles. |
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For, since the straight line AD meets the parallel lines DE, AB, it makes angles BAD, ADE equal to two right angles. |
(I. 29) |
But angle BAD is a right angle; |
(Construction) |
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therefore angle ADE is also a right angle. |
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And in a parallelogram the opposite angles are equal; |
(I. 34) |
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therefore each of the opposite angles DEB, EBA is also a right angle. |
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Therefore ADEB has all its angles right angles. |
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And we proved that all the sides were equal. |
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Therefore ADEB is a square, |
(Definition 7) |
and we have drawn it on the given straight line AB. Q.E.F. |
Please "turn" the page and do some Problems.
or
Continue on to the next proposition.
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