Posts tagged 'polygon'

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G.88 rectangle on bisected and extended segment

If a segment is bisected and a segment is added to it in a line, the rectangle contained by the whole with the added segment and the added segment together with the square on the half equals the square on the segment made up of the half and the added segment.
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G.227 segments proportional rectangle

If four segments be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and, if the rectangle contained by the extremes equals the rectangle contained by the means, the four segments is proportional.
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G.232 similar polygons divided

Similar polygons are divided into similar triangles, and into triangles equal in multitude and in the same ratio as the wholes, and the polygon has to the polygon a ratio duplicate of that which the corresponding side has to the corresponding side.
K A B L C D M E F N G H S Q R O P

G.234 segments proportional polygons

If four segments be proportional, the polygons similar and similarly described upon them is proportional; and, if the polygons similar and similarly described upon them be proportional, the segments will themselves also be proportional.
D E G F H A C K B L M N

G.239 parallelograms applied segment

Of all the parallelograms applied to the same segment and deficient by parallelogrammic figures similar and similarly situated to that described on the half of the segment, that parallelogram is greatest which is applied to the half of the segment and is similar to the defect.
H G P F T O V Q U W R A E S B C D L M K N

G.240 segment apply parallelogram

To a given segment to apply a parallelogram equal to a given polygon and deficient by a parallelogrammic figure similar to a given one: thus the given polygon must not be greater than the parallelogram described on the half of the segment and similar to the defect.
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G.415 segments incommensurable square

If two segments incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum of the squares on them, be added together, the whole segment is irrational; and let it be called the side of the sum of two medial areas.
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G.457 medial segment there

If from a medial segment there be subtracted a medial segment which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, the remainder is irrational; and let it be called a second apotome of a medial segment.
D F G I H E A C B

G.460 segment there subtracted

If from a segment there be subtracted a segment which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called that which produces with a medial area a medial whole.
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G.466 segment produces medial

To a segment which produces with a medial area a medial whole only one segment can be annexed which is incommensurable in square with the whole segment and which with the whole segment makes the sum of the squares on them medial and twice the rectangle contained by them both medial and also incommensurable with the sum of the squares on them.
A D E F G C B H I K L N P S O Q U V W R T M

G.479 area contained rational

If an area be contained by a rational segment and a third apotome, the “side” of the area is a second apotome of a medial segment.
H N O M L A B C K D G F E

G.546 segments proportional parallelepipedal

If three segments be proportional, the parallelepipedal solid formed out of the three is equal to the parallelepipedal solid on the mean which is equilateral, but equiangular with the aforesaid solid.
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G.547 segments proportional parallelepipedal

If four segments be proportional, the parallelepipedal solids on them which are similar and similarly described is proportional; and, if the parallelepipedal solids on them which are similar and similarly described be proportional, the segments will themselves also be proportional.
L A T W B K D U V C N E S P H M F R Q G O

G.570 similar cones cylinders

Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.
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G.586 equilateral equiangular pentagon

If in an equilateral and equiangular pentagon segments subtend two angles taken in order, they cut one another in extreme and mean ratio, and their greater segments are equal to the side of the pentagon.
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G.588 equilateral pentagon inscribed

If an equilateral pentagon be inscribed in a circle, the square on the side of the pentagon is equal to the squares on the side of the hexagon and on that of the decagon inscribed in the same circle.
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G.589 circle diameter rational

If in a circle which has its diameter rational an equilateral pentagon be inscribed, the side of the pentagon is the irrational segment called minor.
A B C D E H K N F G L M

G.595 construct a cube and comprehend it in a

Construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple of the square on the side of the cube.
R Q L F E M Z P W A' V X G K S U N O H T A C D B

G.597 construct an icosahedron and comprehend it in a

Construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the icosahedron is the irrational segment called minor.
U X V M E F R N P S O B H C T W Z G Q K A L D

G.600 construct a dodecahedron and comprehend it in a

Construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures, and to prove that the side of the dodecahedron is the irrational segment called apotome.