## The First Six and the Eleventh and Twelfth Books of Euclid's Elements: With Notes and Illustrations, and an Appendix in Five Books |

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Side 13

OTHERWISE Let the straight line AF divide the angle BAC into two

Then , in the triangles BAF , CAF , the side BA is

common ; therefore the two sides BA , AF are

OTHERWISE Let the straight line AF divide the angle BAC into two

**equal**parts .Then , in the triangles BAF , CAF , the side BA is

**equal**( hyp . ) to CA , and AF iscommon ; therefore the two sides BA , AF are

**equal**to the two CA , AF , each to ... Side 27

has been shown , that it is not

. Wherefore , if two triangles , & c . PROP . XXVI . THEOR . D F If two angles of

one triangle be

...

has been shown , that it is not

**equal**to it : therefore , the angle A is greater than D. Wherefore , if two triangles , & c . PROP . XXVI . THEOR . D F If two angles of

one triangle be

**equal**to two angles of another , each to each , and if a side of the...

Side 79

chords are

extremities of the chords , it will ( I. 29. ) make

therefore the arcs on which these stand are

chords are

**equal**. For , if a straight line be drawn transversely , joining twoextremities of the chords , it will ( I. 29. ) make

**equal**angles with the chords ; andtherefore the arcs on which these stand are

**equal**. PROP . XXVII . THEOR . In**equal**... Side 80

Then , because the circles are

BC is

ELF . Byt ( III . 26. )

Then , because the circles are

**equal**, BK , KC are**equal**to EL , LF ; and the baseBC is

**equal**( byp . ) to the base EF ; therefore ( 1. 8. ) the angle BKC is**equal**toELF . Byt ( III . 26. )

**equal**angles stand upon**equal**arcs , when they are at the ... Side 143

D angular , and the

the angle ABC

in the straight line EF , make ( I. 23. ) the angle FEG

...

D angular , and the

**equal**angles are opposite to the homologous sides , viz . ,the angle ABC

**equal**to DEF , BCA to EFD , and BAC to EDF . At the points E , F ,in the straight line EF , make ( I. 23. ) the angle FEG

**equal**to B , and EFG**equal**to...

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The First Six and the Eleventh and Twelfth Books of Euclid's Elements: With ... Euclid Ingen forhåndsvisning tilgjengelig - 2016 |

### Vanlige uttrykk og setninger

ABCD altitude base bisected called centre chord circle circumference coincide common cone consequently const construction contained continual cylinder demonstrated describe diagonal diameter difference divided double draw equal equal angles extremities figure fore four fourth given given circle given point given straight line greater half Hence inscribed join less magnitudes manner means meet method multiple opposite parallel parallelepiped parallelogram pass perpendicular plane polygon prism PROB produced proof PROP proportional proposition proved pyramid radius ratio reason rectangle remaining respectively right angles Schol segments semicircle shown sides similar square straight line taken THEOR third touching triangle triangle ABC twice vertical wherefore whole

### Populære avsnitt

Side 94 - 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 53 - If a straight line be divided into any two parts, the squares of the whole line and of one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts at the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.

Side 143 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Side 4 - A rhombus is that which has all its sides equal, but its angles are not right angles.

Side 57 - 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 on the other part.

Side 138 - IF a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally; and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle...

Side 43 - In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.

Side 32 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Side 40 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Side 36 - PARALLELOGRAMS upon the same base, and between the same parallels, are equal to one another...