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

### Inni boken

Side 13

To bisect a given finite straight line , that is , to

References — Prop . ... Conclusion . — Therefore the straight line AB is

into two equal parts in the point D . Q . E . F . PROPOSITION 11 . - PROBLEM .

To bisect a given finite straight line , that is , to

**divide**it into two equal parts . (References — Prop . ... Conclusion . — Therefore the straight line AB is

**divided**into two equal parts in the point D . Q . E . F . PROPOSITION 11 . - PROBLEM .

Side 51

If there be two straight lines , one of which is

the rectangle contained by the two straight lines is equal to the rectangles

contained by the undivided line , and the several parts of the

References ...

If there be two straight lines , one of which is

**divided**into any number of parts ,the rectangle contained by the two straight lines is equal to the rectangles

contained by the undivided line , and the several parts of the

**divided**line . (References ...

Side 53

THEOREM . If a straight line be

contained by the whole and one of the parts , is equal to the square on that part ,

together with the rectangle contained by the two parts . ( References — Prop . I .

31 , 46 . ) ...

THEOREM . If a straight line be

**divided**into any two parts , the rectanglecontained by the whole and one of the parts , is equal to the square on that part ,

together with the rectangle contained by the two parts . ( References — Prop . I .

31 , 46 . ) ...

Side 63

PROBLEM , To

contained by the whole and one of the parts shall be ... Produce GH to K . Then

AB shail be

AH .

PROBLEM , To

**divide**a given straight line into two parts , so that the rectanglecontained by the whole and one of the parts shall be ... Produce GH to K . Then

AB shail be

**divided**in II , so that the rectangle AB BH is equal to the square onAH .

Side 68

them may be the greatest possible . ... contained by the sum and difference of the

segments into which the base is

angle .

**Divide**a given straight line into two parts , so that the rectangle contained bythem may be the greatest possible . ... contained by the sum and difference of the

segments into which the base is

**divided**by a perpendicular from the verticalangle .

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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 Conclusion.—Therefore const Construction Construction.-1 Demonstration Demonstration.-1 describe diameter distance divided double draw drawn equal exterior angle extremities fall figure four given circle given point given straight line greater half Hypothesis Hypothesis.—Let impossible inscribed join less manner meet opposite angles parallel parallelogram pass perpendicular produced proved Q. E. D. PROPOSITION reason rectangle contained rectilineal figure References-Prop right angles segment semicircle Sequence.—The shown sides square on AC straight line 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.