## Euclid's Elements of geometry, books i. ii. iii. iv |

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Side 3

A

A

**parallelogram**is a four - sided figure , of which the opposite sides are parallel ; and the diagonal is the straight line joining two of its opposite angles . All other four - sided figures are called trapeziums . POSTULATES . 1. Side 34

The opposite sides and angles of a

The opposite sides and angles of a

**parallelogram**are equal to one another , and the diagonal bisects it , that is , divides it into two equal parts . ( References - Prop . I. 26 , 29 ; ax . 2. ) Hypothesis . Let ACDB be a**parallelogram**... Side 35

Let the

Let the

**parallelograms**ABCD , EBCF , be on the same base BC , and between the same parallels AF , BC . Sequence . The**parallelogram**ABCD shall be equal to the**parallelogram**EBCF . Case I. - If the sides AD , DF , of the**parallelograms**... Side 36

Let ABCD , EFGH , be

Let ABCD , EFGH , be

**parallelograms**on equal bases BC , FG , and between the same parallels AH , BG . Sequence . The**parallelogram**ABCD shall be equal to the**parallelogram**EFGH . Construction . - Join BE , H CH . ―――― Demonstration . Side 37

Then each of the figures EBCA , DBCF , is a

Then each of the figures EBCA , DBCF , is a

**parallelogram**( def . ) , and they are equal to one another , because they are on the same base BC , and between the same parallels BC , EF . ( I. 35. ) B 2. And the triangle ABC is half of ...### Hva folk mener - Skriv en omtale

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ABCD angle ABC angle BAC angle BCD angle equal assumed base base BC BC is equal bisected BOOK centre circle ABC circumference coincide common Conclusion const Construction Construction.-1 Demonstration Demonstration.-1 describe diameter distance divided double draw drawn equal equilateral exterior angle extremities fall figure four given circle given point given straight line Given.-Let greater half Hypothesis Hypothesis.-Let inscribed join less manner meet opposite angle parallel parallelogram pass pentagon perpendicular produced proved Q. E. D. PROPOSITION reason rectangle AB BC rectangle contained References-Prop regular right angles segment semicircle Sequence shown sides Sought square on AC Take third touches the circle triangle ABC twice the rectangle whole

### Populære avsnitt

Side 25 - If two triangles have two angles of the one equal to two angles of the other, each to each ; and one side equal to one side, viz.

Side 2 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

Side 99 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Side 4 - If a straight line meets two straight lines, so as to " make the two interior angles on the same side of it taken " together less than two right angles...

Side 66 - ... the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse...

Side 65 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.

Side 32 - F, which is the common vertex of the triangles ; that is, together with four right angles. Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Side 58 - If a straight line be bisected, and produced to any point ; the rectangle contained by the whole line thus produced, and the part of it produced...

Side 88 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...

Side 33 - The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.