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In a right-angled triangle the square of the hypotenuse, or side subtending the right angle, is equal to the squares of the sides which contain the right angle.

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If the square of one side of a triangle is equal to the squares of the other two sides, the angle subtended by that side is a right angle.

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BOOK II.

PROP. I. THEOR.

The rectangle contained by two straight lines, of which one is divided into any number of parts, is equal to the sum of the rectangles contained by the undivided line and the several parts of the divided line.

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If a straight line be divided into any two parts, the square of the whole line is equal to the sum of the rectangles contained by the whole line and each of its parts.

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If a straight line be divided into any two parts, the rectangle contained by the whole line and either of its parts, is equal to the square of that part, together with the rectangle under the parts.

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If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the parts, together with twice the rectangle contained by the parts.

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COR. Thus it appears that the parallelograms about the diagonal of a square are squares. It is also evident that when a line is bisected, the rectangle contained by the parts is the square of half the line, and therefore that the square of the whole line is equal to four times the square of the half.

PROP. V. THEOR.

If a straight line be bisected, and be also divided unequally, the rectangle contained by the unequal parts, together with the square of the part intermediate between the points of section, is equal to the square of half the line.

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If a straight line be bisected and produced to any point, the rectangle contained by the whole line produced and the produced part, together with the square of half the line, is equal to the square of the line made up of the half and the produced part.

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PROP. VII. THEOR.

If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole line and that part, together with the square of the other part.

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If a straight line be divided into any two parts, the square of the sum of the whole line and one of its parts is equal to four times the rectangle contained by the whole line and that part, together with the square of the other part.

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