## 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 ,

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 if it did meet the circle in two points it would fall ... 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 ... Side 88

Because DE

Because DE

**touches the circle**ABC , and FC is drawn from the assumed centre to the point of contact ; 2. Therefore FC is perpendicular to DE ; ( III . 18. ) 3. Therefore FCE is a right angle . 4. But the angle ACE is also a right angle ... Side 99

Because the straight line EF

Because the straight line EF

**touches the circle**ABCD at the point B ( hyp . ) , and BA is drawn at right angles to the tangent from the point of contact B , ( const . ) 2. The centre of the circle is in BA . ( III . 19. ) 3. Side 101

And the circle described from the centre G , at the distance GA , will therefore pass through the point B. 6. Let this circle be described ; and let it be AHB . 7. ... Therefore AD

And the circle described from the centre G , at the distance GA , will therefore pass through the point B. 6. Let this circle be described ; and let it be AHB . 7. ... Therefore AD

**touches the circle**. ( III . 16 , cor . ) 9.### Hva folk mener - Skriv en omtale

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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 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.