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14 Cards in this Set
- Front
- Back
- 3rd side (hint)
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Proposition 2.1.1 |
If there are two straight lines and one of them is cut into any number of pieces whatsoever then the rectangle contained by two straight lines is equal to the sum of the rectangles contained by uncut straight line and every one of the pieces of the cut straight line. |
Let A and BC be two straight lines and let BC be cut at random at points D and E I say that rectangle contained by A and BC is equal to the rectangles contained by A and BD by A and DE and by A and EC. |
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Proposition 2.2. |
If a straight line cut at random then the sum of the rectangles contained by whole straight line and each of the pieces of the straight line is equal to the square on the whole. |
For let straight line have been cut at random at point C I say that rectangle contained by AB and BC plus rectangle contained by BA and AC is equal to the square on AB |
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Proposition 2.3 |
If a straight line is cut at random then the rectangle contained by whole and one of the pieces of the straight line is equal to the rectangle contained by both the pieces and the square on the aforementioned piece |
For let the straight line AB have been cut at random at point C I say that rectangle contained by AB and BC is equal to the rectangle contained by AB and CB plus the square on the BC. |
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Proposition 2.4 |
If a straight line cut at random then the square on the whole straight line is equal to the sum of the squares on the pieces of the straight line and twice the rectangle contained by the pieces. |
For let the straight line AB have been cut at random at point C I say that square on AB is equal to the sum of the squares on AC and CB and twice the rectangle contained by AC and BC. |
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Proposition 2.5 |
If a straight line is cut into equal and unequal pieces then the rectangle contained by the unequal pieces of the whole straight line plus the square on the difference between the equal and unequal pieces is equal to the square on the half of the straight line |
For let any straight line AB have been cut equally at C and unequally at D I say that rectangle contained by AD and DB plus square on CD is equal to square on the CB |
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Proposition 2.6 |
If a straight line is cut into half and any straight line added to it straight on then the rectangle contained by the whole straight line with the straight line being added, and the straight line having being added plus the square on the half of the original straight line is equal to square on the sum of half of the original straight line and the straight line having being added. |
For let any straight line AB have been cut in half at point C and let any straight line BD have been added to it straight on I say that rectangle contained by AD and DB plus the square on the CB is equal to the square on the CD. |
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Proposition 2.7 |
If a straight line is cut at random then the sum of the squares on the whole straight line and one of the pieces of the straight line is equal to twice the rectangle contained by the whole and the said piece and the square on the remaining piece. |
For let straight line have been cut at random point C I say that sum of the squares on AB and BC is equal to twice the rectangle contained by AB and BC and square on the CA. |
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Proposition 2.8 |
If the straight line is cut at random then four times the rectangle contained by the whole and one of the pieces of the straight line plus the square on the remaining piece is equal to square described on whole and the former piece as on one complete straight line. |
For let any straight line AB have been cut at random at point C I say that four times the rectangle contained by AB and BC plus the square on the AC is equal to the square described on AB and BC as on one complete straight line. |
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Proposition 2.9 |
If a straight line is cut into equal and unequal pieces then the sum of the squares on unequal pieces of the whole straight line is double the the sum of the square on half of the straight line and the square on the difference between the equal and the unequal pieces. |
For let any straight line AB have been cut equally at point C and unequally at point D I say that sum of the squares on AD and DB is double the sum of the squares on AC and CD. |
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Proposition 2.10 |
If a straight line is cut into half and any straight line added to it straight on then the sum of the squares on the whole straight line with the straight line having been added and the square on the straight line having been added is double the sum of the square on half of the straight line and the square described on the sum of the half of the straight line and straight line having been added as on one complete straight line. |
For let any straight line AB have been cut in half at point C and let any straight line BD have been added to it straight on I say that the sum of the squares on AD and DB is double the sum of the squares on AC and CD. |
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Proposition 2.11 |
To cut a given straight line such that the rectangle contained by whole straight line and one of the pieces of the straight line is equal to square on the remaining piece. |
Let AB be the given straight line so it is required to cut AB such that the rectangle contained by the whole straight line and one of the pieces of the straight line is equal to the square on the remaining piece |
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Proposition 2.12 |
In obtuse angled triangles the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides around the obtuse angle to which a perpendicular straight line falls and the straight line cut off outside the triangle by the perpendicular straight line towards the obtuse angle. |
Let ABC be an obtuse angled triangle having the angle BAC obtuse and let BD have been drawn from point B perpendicular to CA produced I say that the square on BC is greater than the sum of the squares on AC and AB by twice the rectangle contained by CA and AD. |
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Proposition 2.13 |
In acute angled triangles the square on the side subtending the acute angle is less than the sum of the squares on the sides containing the acute angle by twice the rectangle contained by one of the sides around the acute angle to which a perpendicular straight line falls and the straight line cut off inside the triangle by the perpendicular straight line towards the acute angle |
Let ABC be an acute angled triangle having the angle at point B acute and let AD have been drawn from point A perpendicular to BC I say that the square on AC is less than the sum of the squares on CB and BA by twice the rectangle contained by CB and BD. |
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Proposition 2.14 |
To construct a square equal to the given rectilinear figure. |
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