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

### Inni boken

Resultat 1-5 av 5

Side 5

To

Def . 15 ; ax . 1 ; post . 1 , 3 . ) Given . - Let AB be the given straight line . Sought .

- It is required to

To

**describe**an equilateral triangle on a given finite straight line . * ( References -Def . 15 ; ax . 1 ; post . 1 , 3 . ) Given . - Let AB be the given straight line . Sought .

- It is required to

**describe**an equilateral triangle on AB . Construction . - 1 . Side 6

Upon AB

the straight lines DA , DB , to E and F . ( post . 2 . ) 4 . From the centre B , at the

distance BC ,

...

Upon AB

**describe**the equilateral triangle DAB . ( Book I . prop . 1 . ) 3 . Producethe straight lines DA , DB , to E and F . ( post . 2 . ) 4 . From the centre B , at the

distance BC ,

**describe**the circle CGH , meeting DF in G . ( post . 3 . ) 5 . From the...

Side 50

To

have two sides of the one equal respectively to two sides of the other , and have

likewise the contained angles supplementary , the triangles are equal in area . 42

.

To

**describe**a triangle equal to a given quadrilateral figure . 41 . If two triangleshave two sides of the one equal respectively to two sides of the other , and have

likewise the contained angles supplementary , the triangles are equal in area . 42

.

Side 107

impossible . 2 . With a given radius ,

point , and touching a given straight line . 3 . If two circles cut each other , any two

parallel ...

**Describe**a circle passing through three given points . When will this beimpossible . 2 . With a given radius ,

**describe**a circle passing through a givenpoint , and touching a given straight line . 3 . If two circles cut each other , any two

parallel ...

Side 117

Through E draw EH parallel to AB or DC , and throngh F draw FK parallel to AD

or BC , cutting EH in G . ( I . 31 . ) 3 . From the centre G , at the distance GE , GF ,

or GK ,

.

Through E draw EH parallel to AB or DC , and throngh F draw FK parallel to AD

or BC , cutting EH in G . ( I . 31 . ) 3 . From the centre G , at the distance GE , GF ,

or GK ,

**describe**the circle EFHK . · Then EFHK shall be the circle required . Proof.

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### Vanlige uttrykk og setninger

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 equilateral exterior angle extremities fall figure four given circle given point given rectilineal given straight line greater half Hypothesis.—Let inscribed join less manner meet opposite angle parallel parallelogram pass pentagon perpendicular produced Prop proved Q. E. D. PROPOSITION reason rectangle contained rectilineal figure References—Prop regular right angles segment semicircle shown sides Sought.-It is required 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.