Posts tagged 'diameter'
G.61 diameter of a circle
A diameter of a circle is any segment drawn through the center and terminated in both directions by the arc of the circle; it bisects the circle.
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.66 triangles on the same base in the same parallels
Triangles on the same base and in the same parallels are equal in area.
G.67 triangles on equal bases in the same parallels
Triangles 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.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.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.96 semicircle
A semicircle is the figure contained by a diameter and the arc cut off by it.
G.105 distances to a circle from an eccentric diameter point
From an eccentric point on the diameter of a circle, the greatest segment to the arc passes through the center, the least is the remainder of the diameter, and segments nearer the center are greater than those more remote, with at most two equal segments on opposite sides.
G.106 distances to a circle from an exterior point
From a point outside a circle, of segments falling on the concave arc the greatest passes through the center and nearer ones are greater, while on the convex arc the least lies between the point and the diameter and nearer ones are less, with at most two equal segments.
G.116 diameter is the greatest chord
Of chords in a circle, the diameter is the greatest, and a chord nearer the center is always greater than one more remote.
G.118 perpendicular at extremity of a diameter is tangent
The line drawn at right angles to the diameter of a circle from its extremity falls outside the circle, no other line can be interposed between it and the arc, and the angle of the semicircle is greater than any acute rectilineal angle.
G.119 draw a tangent to a circle from a point
From a given point, construct a line tangent to a given circle.
G.131 equal circles
Equal circles are those the:term:diameters <G.61> of which are equal, or the radii of which are equal.
G.139 Thales’ theorem and angles in circle segments
In a circle, the angle in a semicircle is right, the angle in a greater segment is acute, and the angle in a lesser segment is obtuse.
G.140 alternate segment theorem
If a line touches a circle and a chord is drawn from the point of contact, the angles which the chord makes with the tangent equal the angles in the alternate segments of the circle.
G.141 construct a circle segment on a chord admitting a given angle
On a given segment, construct a segment of a circle admitting an angle equal to a given rectilineal angle.
G.145 converse of the tangent-secant theorem
If from a point outside a circle a secant and a segment to the circle are drawn such that the rectangle on the whole secant and its external part equals the square on the segment, the segment is tangent to the circle.
G.147 fit a chord in a circle equal to a segment
Fit into a given circle a chord equal to a given segment not greater than the diameter of the circle.
G.154 inscribe a circle in a triangle
Inscribe a circle in a given triangle.
G.158 inscribe a square in a circle
Inscribe a square in a given circle.
G.159 circumscribe a square about a circle
Circumscribe a square about a given circle.
G.160 inscribe a circle in a square
Inscribe a circle in a given square.
G.162 construct a golden isosceles triangle
Construct an isosceles triangle having each of the angles at the base double the remaining angle.
G.165 inscribe a circle in a regular pentagon
Inscribe a circle in a given equilateral and equiangular pentagon.
G.167 inscribe a regular hexagon in a circle
Inscribe an equilateral and equiangular hexagon in a given circle.
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.477 area contained rational
If an area be contained by a rational segment and a first apotome, the “side” of the area is an apotome.
G.478 area contained rational
If an area be contained by a rational segment and a second apotome, the “side” of the area is a first apotome of a medial segment.
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.
G.480 area contained rational
If an area be contained by a rational segment and a fourth apotome, the “side” of the area is minor.
G.481 area contained rational
If an area be contained by a rational segment and a fifth apotome, the “side” of the area is a segment which produces with a rational area a medial whole.
G.482 area contained rational
If an area be contained by a rational segment and a sixth apotome, the “side” of the area is a segment which produces with a medial area a medial whole.
G.502 part segment plane
A part of a segment cannot be in the plane of reference and a part in a plane more elevated.
G.528 construct a solid angle out of three plane
Construct a solid angle out of three plane angles two of which, taken together in any manner, are greater than the remaining one: thus the three angles must be less than four right angles.
G.549 sides opposite planes
If the sides of the opposite planes of a cube be bisected, and planes be carried through the points of section, the common section of the planes and the diameter of the cube bisect one another.
G.551 similar polygons inscribed
Similar polygons inscribed in circles are to one another as the squares on the diameters.
G.552 circles squares diameters
Circles are to one another as the squares on the diameters.
G.558 prism triangular base
Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.
G.565 cones cylinders height
:term:`Cones <G.561>` and :term:`cylinders <G.562>` which are of the same :term:`height <G.209>` are to one another as their bases.
G.570 similar cones cylinders
Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.
G.575 sphere
When, the:term:diameter <G.61> of a:term:semicircle <G.96> remaining fixed, the semicircle 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:**sphere** <sphere>.
G.577 spheres center inscribe
Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.
G.578 spheres triplicate ratio
Spheres are to one another in the triplicate ratio of their respective diameters.
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.
G.592 construct pyramid comprehend
To construct a :term:`pyramid <G.553>`, to comprehend it in a given :term:`sphere <G.575>`, and to prove that the :term:`square <G.63>` on the :term:`diameter of the sphere <G.576>` is one and a half times the square on the side of the pyramid.
G.594 construct an octahedron and comprehend it in a
Construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double of the square on the side of the octahedron.
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.
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.
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.
G.601 set out sides
To set out the sides of the five figures and to compare them with one another.