## The Elements of Euclid |

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Resultat 1-5 av 10

Side 82

to the

triangle DEF , and it is inscribed in the circle ABC . Which was to be done . PROP

. III . PROB . About a given circle to describe a triangle equiangular to a given ...

to the

**remaining**angle EDF : wherefore the triangle ABC is equiangu . lar to thetriangle DEF , and it is inscribed in the circle ABC . Which was to be done . PROP

. III . PROB . About a given circle to describe a triangle equiangular to a given ...

Side 121

... then the ratio compounded of the

to E , and E to F , which compounded ratio is the ratio of D to F , is the same with

the ratio of K to M , which is compounded of the

...

... then the ratio compounded of the

**remaining**first ratios , to wit , of the ratios of Dto E , and E to F , which compounded ratio is the ratio of D to F , is the same with

the ratio of K to M , which is compounded of the

**remaining**ratios of K to L , and L...

Side 129

BCA ; wherefore the

32 . 1 . ) , and the triangle ABC is therefore equiangular to the triangle GEF ; and

consequently they have their sides opposite to the в equal angles proportionals ...

BCA ; wherefore the

**remaining**angle BAC is equal to the**remaining**angle EGF (32 . 1 . ) , and the triangle ABC is therefore equiangular to the triangle GEF ; and

consequently they have their sides opposite to the в equal angles proportionals ...

Side 130

and the triangle EDF to the triangle GDF , and the

therefore the angle DFG is equal to the angle DFE , and the angle at G to the

angle at E : but ...

and the triangle EDF to the triangle GDF , and the

**remaining**angles to the**remaining**angles , each to each , which are subtended by the equal sides ;therefore the angle DFG is equal to the angle DFE , and the angle at G to the

angle at E : but ...

Side 169

than CB , and one of them , BD , has been proved equal to BE a part of CB ,

therefore the other DC is greater than the E C

DA is equal to EA , and AC common , but the base DC greater than the base EC ...

than CB , and one of them , BD , has been proved equal to BE a part of CB ,

therefore the other DC is greater than the E C

**remaining**part EC . And becauseDA is equal to EA , and AC common , but the base DC greater than the base EC ...

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added altitude angle ABC angle BAC base BC is given centre circle circle ABCD circumference common cone contained cylinder definition demonstrated described diameter divided double draw drawn equal equal angles equiangular equimultiples Euclid excess fore four fourth given angle given in magnitude given in position given in species given magnitude given ratio given straight line gles greater Greek half join less likewise magnitude manner meet multiple Note opposite parallel parallelogram pass perpendicular plane prism produced PROP proportionals proposition pyramid Q. E. D. PROP reason rectangle rectangle contained remaining right angles segment shown sides similar sine solid sphere square square of BC taken THEOR third triangle ABC wherefore whole

### Populære avsnitt

Side 36 - 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 145 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.

Side 65 - 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 248 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Side 11 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Side 121 - 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 21 - 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 14 - To construct a triangle of which the sides shall be equal to three given straight lines ; but any two whatever of these lines must be greater than the third (20.

Side 80 - 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. either the sides adjacent to the equal...

Side 133 - ... rectilineal figures are to one another in the duplicate ratio of their homologous sides.