Posts tagged 'parallelogram'
G.62 opposite sides, angles, and diagonal of a parallelogram
In a parallelogram, opposite sides and opposite angles are equal, and the diagonal bisects the area.
G.64 parallelograms on the same base in the same parallels
Parallelograms on the same base and in the same parallels are equal in area.
G.65 parallelograms on equal bases in the same parallels
Parallelograms on equal bases and in the same parallels are equal in area.
G.70 parallelogram is double a triangle on the same base
If a parallelogram shares the same base with a triangle and lies in the same parallels, the parallelogram is double the triangle.
G.71 construct a parallelogram equal to a triangle
Construct, in a given angle, a parallelogram equal in area to a given triangle.
G.72 complements of parallelograms about the diagonal are equal
In any parallelogram, the complements of the parallelograms about the diagonal are equal in area.
G.73 apply a parallelogram equal to a triangle on a segment
Apply to a given segment, in a given angle, a parallelogram equal in area to a given triangle.
G.74 construct a parallelogram equal to a polygon
Construct, in a given angle, a parallelogram equal in area to a given polygon.
G.81 contained rectangle
Any rectangular parallelogram is said to be:term:**contained** <contained-rectangle> by the two:term:segments <G.4> containing the:term:right angle <G.33>.
G.86 gnomon
And in any parallelogrammic area let any one whatever of the parallelograms about its:term:diameter <G.61> with the two complements be called a:term:**gnomon** <gnomon>.
G.210 triangles parallelograms height
Triangles and parallelograms which are under the same height are to one another as their bases.
G.225 equal equiangular parallelograms
In equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; and equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal.
G.235 equiangular parallelograms ratio
Equiangular parallelograms have to one another the ratio compounded of the ratios of their sides.
G.236 parallelogram parallelograms diameter
In any parallelogram the parallelograms about the diameter are similar both to the whole and to one another.
G.238 parallelogram there taken
If from a parallelogram there be taken away a parallelogram similar and similarly situated to the whole and having a common angle with it, it is about the same diameter with the whole.
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.
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.
G.241 segment apply parallelogram
To a given segment to apply a parallelogram equal to a given polygon and exceeding by a parallelogrammic figure similar to a given one.
G.389 there unequal segments
If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are commensurable in length, then the square on the greater is greater than the square on the less by the square on a segment commensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment commensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it will divide it into parts which are commensurable in length.
G.390 there unequal segments
If there be two unequal segments, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are incommensurable, the square on the greater is greater than the square on the less by the square on a segment incommensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a segment incommensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable.
G.530 solid contained parallel
If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.
G.537 prism
A:term:**prism** <prism> is a:term:solid figure <G.529> contained by:term:planes <G.7> two of which, namely those which are opposite, are equal, similar and:term:parallel <G.52>, while the rest are parallelograms.
G.550 there prisms equal
If there be two prisms of equal height, and one have a parallelogram as base and the other a triangle, and if the parallelogram be double of the triangle, the prisms is equal.
G.562 cylinder
When, one side of those about the:term:right angle <G.33> in a:term:rectangular parallelogram <G.81> remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the:term:figure <G.9> so comprehended is a:term:**cylinder** <cylinder>.
G.567 axis of a cylinder
The:term:**axis of the cylinder** <axis-of-cylinder> is the:term:segment <G.4> which remains fixed and about which the parallelogram is turned.