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

### Inni boken

Side 154

Again , because the quadrilaterals ABHF , CDGE , and the

are

FH : HL :: EG : GK . In like manner , it may be shown , that AB : BL :: CD : DK : and

...

Again , because the quadrilaterals ABHF , CDGE , and the

**triangles**BLH , DKGare

**similar**, as FH : HB :: EG : GD , and HB : HL :: GD : GK ; therefore , ex æquo ,FH : HL :: EG : GK . In like manner , it may be shown , that AB : BL :: CD : DK : and

...

Side 155

the triangle ABC has to ABG the duplicate ratio of that which BC has to EF . But

the triangle ABG is equal to DEF ; wherefore also the triangle ABC has to DEF the

duplicate ratio of that which BC has to EF . * Therefore ,

the triangle ABC has to ABG the duplicate ratio of that which BC has to EF . But

the triangle ABG is equal to DEF ; wherefore also the triangle ABC has to DEF the

duplicate ratio of that which BC has to EF . * Therefore ,

**similar triangles**, & c . Side 156

similar . In the same manner , the triangles ECD , LHK may be shown to be

similar . Therefore the similar polygons AD , FK are divided into the same number

of

FGL ...

similar . In the same manner , the triangles ECD , LHK may be shown to be

similar . Therefore the similar polygons AD , FK are divided into the same number

of

**similar triangles**. 2. Because the triangles ABE , FGL are similar , ABE has toFGL ...

Side 345

The surfaces of two similar polyhedrons may be divided into the same number of

XI . def . 8. ) of similar bodies bounded by planes , if the sides or faces of the ...

The surfaces of two similar polyhedrons may be divided into the same number of

**similar triangles**, similarly situated . This follows immediately from the definition (XI . def . 8. ) of similar bodies bounded by planes , if the sides or faces of the ...

Side 347

into the same number of

through any two corresponding solid angles , A , a , and through the sides of all

these triangles , except those forming h d c D G the solid angles , A , a , will ...

into the same number of

**similar triangles**, similarly situated : then planes passingthrough any two corresponding solid angles , A , a , and through the sides of all

these triangles , except those forming h d c D G the solid angles , A , a , will ...

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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...