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

### Inni boken

Side 90

The opposite angles of any quadrilateral figure

equal to two right angles . ( References — Prop . I . 32 ; III . 21 . ) Hypothesis . —

Let ABCD be a quadrilateral figure

The opposite angles of any quadrilateral figure

**inscribed**in a circle , are togetherequal to two right angles . ( References — Prop . I . 32 ; III . 21 . ) Hypothesis . —

Let ABCD be a quadrilateral figure

**inscribed**in the circle ABCD . Sequence . Side 109

A rectilineal figure is said to be

the angles of the

about ...

A rectilineal figure is said to be

**inscribed**in another rectilineal figure , when allthe angles of the

**inscribed**figure are upon the sides of the figure in which it is**inscribed**, each upon each . 2 . In like manner , a figure is said to be describedabout ...

Side 126

Therefore the hexagon is equiangular ; and it was shown to be equilateral .

Conclusion . Therefore , a regular hexagon has been

ABCDEF . Q . E . F . Corollary . - 1 . From this it is manifest that the side of the

hexagon is ...

Therefore the hexagon is equiangular ; and it was shown to be equilateral .

Conclusion . Therefore , a regular hexagon has been

**inscribed**in the circleABCDEF . Q . E . F . Corollary . - 1 . From this it is manifest that the side of the

hexagon is ...

Side 127

Let AC be the side of an equilateral triangle

AB the side of a regular pentagon

arc BC in E ( III . 30 ) , and join BE , EC . 3 . Draw straight lines equal to BE or EC

...

Let AC be the side of an equilateral triangle

**inscribed**in the circle ( IV . 2 ) , andAB the side of a regular pentagon

**inscribed**in the same . ( IV . 11 . ) 2 . Bisect thearc BC in E ( III . 30 ) , and join BE , EC . 3 . Draw straight lines equal to BE or EC

...

Side 128

If a triangle be

equiangular to it , in the other circle , and having its sides respectively

perpendicular to those of the first triangle . 3 . The square of the side of an

equilateral triangle ...

If a triangle be

**inscribed**in one of two concentric circles , inscribe anotherequiangular to it , in the other circle , and having its sides respectively

perpendicular to those of the first triangle . 3 . The square of the side of an

equilateral triangle ...

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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 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.