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

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Side 86

which is drawn at right angles to the diameter of a circle , from the extremity of it ,

it did meet the circle in two points it would fall within it ( III . 2 ) . Also it is evident ...

which is drawn at right angles to the diameter of a circle , from the extremity of it ,

**touches the circle**( III . def . 2 ) ; and that it touches it only in one point , because ifit did meet the circle in two points it would fall within it ( III . 2 ) . Also it is evident ...

Side 87

Next , let the given point be in the circumference of the circle , as the point D . 12 .

Draw DE to the centre E , and DF at right angles to DE ; 13 . Then DF

Next , let the given point be in the circumference of the circle , as the point D . 12 .

Draw DE to the centre E , and DF at right angles to DE ; 13 . Then DF

**touches the****circle**. ( III . 16 , cor . ) Conclusion . — Therefore from the given points A and D ... Side 101

Let this circle be described ; and let it be AHB . 7 . Because from the point A , the

extremity of the diameter AE , AD is drawn at right angles to AE ; ( const . ) 8 .

Therefore AD

...

Let this circle be described ; and let it be AHB . 7 . Because from the point A , the

extremity of the diameter AE , AD is drawn at right angles to AE ; ( const . ) 8 .

Therefore AD

**touches the circle**. ( III . 16 , cor . ) 9 . Because AB is drawn from the...

Side 106

of which DCA cuts the circle , and DB meets it ; and let the rectangle AD . DC , be

equal to the square on DB . Sequence . - Then DB shall touch the circle .

Construction . — 1 . Draw the straight line DE ,

) 2 .

of which DCA cuts the circle , and DB meets it ; and let the rectangle AD . DC , be

equal to the square on DB . Sequence . - Then DB shall touch the circle .

Construction . — 1 . Draw the straight line DE ,

**touching the circle**ABC ; ( III . 17 .) 2 .

Side 117

And because the angles at the points E , F , H , K , are right angles ( I . 29 ) ; and

that the straight line which is drawn from the extremity of a diameter , at right

angles to it ,

lines ...

And because the angles at the points E , F , H , K , are right angles ( I . 29 ) ; and

that the straight line which is drawn from the extremity of a diameter , at right

angles to it ,

**touches the circle**; ( III . 16 , cor . ) 8 . Therefore each of the straightlines ...

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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 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.