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

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Side 19

Any two of its angles together shall be less than two right angles . Construction .

Produce BC to D . Demonstration . - 1 . Because ACD is the exterior angle of the

triangle ABC , it is greater than the interior and opposite

Any two of its angles together shall be less than two right angles . Construction .

Produce BC to D . Demonstration . - 1 . Because ACD is the exterior angle of the

triangle ABC , it is greater than the interior and opposite

**angle ABC**. ( I . 16 . ) 2 . Side 20

Much more then is the

Therefore , the greater side , & c . Q . E . D . PROPOSITION 19 . - THEOREM . The

greater angle of every triangle is subtended by the greater side , or has the ...

Much more then is the

**angle ABC**greater than the angle ACB . Conclusion . —Therefore , the greater side , & c . Q . E . D . PROPOSITION 19 . - THEOREM . The

greater angle of every triangle is subtended by the greater side , or has the ...

Side 90

Because the three

( 1 . 32 . ) 2 . The three

are together equal to two right

Because the three

**angles**of every triangle are together equal to two right**angles**,( 1 . 32 . ) 2 . The three

**angles**of the triangle CAB , namely , CAB , ACB ,**ABC**,are together equal to two right

**angles**. 3 . But the**angle**ČAB is equal to the ... Side 98

The angle EAB is equal to the angle EBA ; ( I . 5 . ) 3 . And , because EA is equal

to EC , 4 . The angle EAC is equal to the angle ECA ; 5 . Therefore the whole

angle BAC is equal to the two

exterior ...

The angle EAB is equal to the angle EBA ; ( I . 5 . ) 3 . And , because EA is equal

to EC , 4 . The angle EAC is equal to the angle ECA ; 5 . Therefore the whole

angle BAC is equal to the two

**angles ABC**, ACB . ( ax . 2 . ) 6 . But FAC , theexterior ...

Side 111

It is required to inscribe in the circle

DEF . Construction . — 1 . Draw the straight line GAH touching the circle in the

point A . ( III . 17 . ) 2 . At the point A , in the straight line AH , make the

...

It is required to inscribe in the circle

**ABC**a triangle equiangular to the triangleDEF . Construction . — 1 . Draw the straight line GAH touching the circle in the

point A . ( III . 17 . ) 2 . At the point A , in the straight line AH , make the

**angle**HAC...

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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.