Posts tagged 'line'
G.2 line
A line [ A B ] is a projection of a relationship of two points A, B.
G.3 extremities of a segment
The extremities of a segment / A B / are points A, B.
G.4 segment
A segment / A B / is a portion of a line [ A B ] between two points A, B on the line.
G.6 extremities of a surface
The extremities of a surface are lines or arcs.
G.22 plane angle
A plane angle is the inclination to one another of two lines or arcs in a plane that meet and do not lie on the same line.
G.30 rectilineal angle
A rectilineal angle is contained by two segments or lines.
G.35 construct a perpendicular from a point to a line
Construct a perpendicular to a given line from a given point not on it.
G.45 triangle inequality
In any triangle, the sum of any two sides is greater than the remaining side.
G.46 interior segments of a triangle
Segments constructed from the endpoints of one side of a triangle to an interior point have a lesser sum but contain a greater angle than the other two sides.
G.47 construct a triangle from three segments
Construct a triangle from three segments equal to three given segments satisfying the triangle inequality.
G.48 construct an angle equal to a given angle
Construct on a given line at a given point an angle equal to a given angle.
G.58 construct a line parallel to a given line
Construct through a given point a line parallel to a given line.
G.59 exterior angle and angle sum of a triangle
In any triangle, an exterior angle equals the sum of the two opposite interior angles, and the three interior angles sum to two right angles.
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.94 law of cosines for obtuse triangles
In obtuse-angled triangles, the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by one of those sides and the projection of the other upon it.
G.97 construct a square equal to a polygon
Construct a square equal in area to a given rectilineal figure.
G.114 equal chords are equidistant from the center
In a circle, chords are equal if and only if they are equally distant from the center.
G.120 radius to point of contact is perpendicular to tangent
If a line touches a circle, the segment joined from the center to the point of contact is perpendicular to the tangent.
G.136 equal arcs are subtended by equal chords
In equal circles, equal arcs are subtended by equal chords.
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.144 tangent-secant theorem
If from a point outside a circle a secant and a tangent are drawn, the rectangle contained by the whole secant and its external segment equals the square on the tangent.
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.160 inscribe a circle in a square
Inscribe a circle in a given square.
G.163 inscribe a regular pentagon in a circle
Inscribe an equilateral and equiangular pentagon in a given circle.
G.165 inscribe a circle in a regular pentagon
Inscribe a circle in a given equilateral and equiangular pentagon.
G.166 circumscribe a circle about a regular pentagon
Circumscribe a circle about 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.211 segment drawn parallel
If a segment be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of the triangle be cut proportionally, the line joining the points of section is parallel to the remaining side of the triangle.
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.242 extreme and mean ratio
A segment is said to have been cut in extreme and mean ratio when, as the whole:term:line <G.2> is to the greater segment, so is the greater to the less.
G.504 planes cut common
If two planes cut one another, their common section is a segment.
G.524 solid angle
A:term:**solid angle** <solid-angle> is the inclination constituted by more than two:term:lines <G.2> which meet one another and are not in the same:term:surface <G.533>, towards all the lines. Otherwise: A:term:**solid angle** <solid-angle> is that which is contained by more than two:term:plane angles <G.22> which are not in the same:term:plane <G.7> and are constructed to one:term:point <G.1>.
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.