Posts tagged 'circle'
G.10 circle
A circle ( A B ) is a plane figure contained by one arc defined by a center A and a point B establishing its defining radius / A B /, such that all segments / A B /, / A C /, / A D / falling upon the arc from A are equal.
G.11 center of a circle
The point A within the circle ( A B ) from which all segments / A B /, / A C / to the arc are equal is called the center of the circle.
G.15 construct a circle
Construct a circle ( A B ) with center A and defining radius point B.
G.17 construct an equilateral triangle
Construct an equilateral triangle on a given segment.
G.20 transfer a segment to a point
Construct at a given point a segment equal to a given segment.
G.21 cut off an equal segment
Cut off from the greater of two unequal segments a segment equal to the less.
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.47 construct a triangle from three segments
Construct a triangle from three segments equal to three given segments satisfying the triangle inequality.
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.96 semicircle
A semicircle is the figure contained by a diameter and the arc cut off by it.
G.97 construct a square equal to a polygon
Construct a square equal in area to a given rectilineal figure.
G.98 find the center of a circle
Find the center of a given circle.
G.99 chord of a circle lies within the circle
The segment joining any two points on the arc of a circle falls within the circle.
G.100 diameter bisects chord iff perpendicular
A segment through the center of a circle bisects a chord not through the center if and only if it cuts the chord at right angles.
G.101 intersecting chords not through the center do not bisect
In a circle, two intersecting chords that do not both pass through the center do not bisect one another.
G.102 intersecting circles have distinct centers
Two intersecting circles do not have the same center.
G.104 touching circles have distinct centers
Two circles that touch one another do not have the same center.
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.107 point with three equal segments to a circle is the center
A point within a circle from which more than two equal segments fall on the circle is the center of the circle.
G.108 two circles intersect in at most two points
A circle does not intersect another circle at more than two points.
G.109 line of centers of internally touching circles
If two circles touch one another internally, the line joining their centers passes through the point of contact.
G.110 line of centers of externally touching circles
If two circles touch one another externally, the segment joining their centers passes through the point of contact.
G.111 line tangent to a circle
A:term:segment <G.4> is said to:term:**touch a circle** <tangent-to-circle> which, meeting the:term:circle <G.10> and being produced, does not cut the circle.
G.112 two circles touch at most at one point
A circle does not touch another circle at more than one point, whether internally or externally.
G.113 chords equally distant from the center
In a:term:circle <G.10>segments are said to be:term:**equally distant from the center** <equally-distant-from-center> when the:term:perpendiculars <G.33> drawn to them from the:term:center <center> are equal.
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.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.117 angle of a circle segment
An:term:**angle of a segment** <angle-of-segment> is that contained by a:term:segment <G.4> and a arc of a:term:circle <G.10>.
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.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.121 perpendicular to tangent at contact passes through center
If a line touches a circle, the line drawn at right angles to the tangent from the point of contact passes through the center of the circle.
G.122 sector of a circle
A:term:**sector of a circle** <sector-of-circle> is the:term:figure <G.9> which, when an:term:angle <G.22> is constructed at the:term:center <center> of the:term:circle <G.10>, is contained by the:term:segments <G.4> containing the angle and the arc cut off by them.
G.123 inscribed angle theorem
In a circle, the angle at the center is double the angle at the circumference when the angles have the same arc as base.
G.124 angles in the same circle segment are equal
In a circle, angles in the same segment are equal to one another.
G.125 opposite angles of a cyclic quadrilateral are supplementary
The opposite angles of any quadrilateral inscribed in a circle sum to two right angles.
G.126 segment of a circle
A:term:**segment of a circle** <segment-of-circle> is the:term:figure <G.9> contained by a:term:segment <G.4> and a arc of a:term:circle <G.10>.
G.128 no two similar unequal circle segments on the same chord
On the same segment and on the same side of it, there cannot be constructed two similar and unequal segments of circles.
G.129 similar circle segments on equal chords are equal
Similar segments of circles on equal segments are equal to one another.
G.130 complete a circle from a given segment
Given a segment of a circle, complete the full circle of which it is a segment.
G.133 equal angles stand on equal arcs
In equal circles, equal angles stand on equal arcs, whether they stand at the centers or at the circumferences.
G.134 angles standing on equal arcs are equal
In equal circles, angles standing on equal arcs are equal to one another, whether they stand at the centers or at the circumferences.
G.135 equal chords cut off equal arcs
In equal circles, equal chords cut off equal arcs, the greater equal to the greater and the less to the less.
G.136 equal arcs are subtended by equal chords
In equal circles, equal arcs are subtended by equal chords.
G.137 bisect an arc
Bisect a given arc of a circle.
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.142 cut off a circle segment admitting a given angle
From a given circle, cut off a segment admitting an angle equal to a given rectilineal angle.
G.143 intersecting chords theorem
If two chords in a circle intersect one another, the rectangle contained by the segments of one equals the rectangle contained by the segments of the other.
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.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.146 segment fitted into a circle
A:term:segment <G.4> is said to be:term:**fitted into a circle** <fitted-into-circle> when its:term:extremities <G.6> are on the arc of the:term:circle <G.10>.
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.148 polygon inscribed in a circle
A:term:polygon <G.12> is said to be:term:**inscribed in a circle** <inscribed-in-circle> when each:term:angle <G.22> of the inscribed:term:figure <G.9> lies on the arc of the:term:circle <G.10>.
G.149 inscribe a triangle in a circle equiangular with a triangle
Inscribe in a given circle a triangle equiangular with a given triangle.
G.150 polygon circumscribed about a circle
A:term:polygon <G.12> is said to be:term:**circumscribed about a circle** <circumscribed-about-circle>, when each side of the circumscribed:term:figure <G.9> touches the arc of the:term:circle <G.10>.
G.151 circumscribe a triangle about a circle equiangular with a triangle
Circumscribe about a given circle a triangle equiangular with a given triangle.
G.153 circle inscribed in a polygon
Similarly a:term:circle <G.10> is said to be:term:**inscribed in a figure** <circle-inscribed-in-figure> when the arc of the circle touches each side of the:term:figure <G.9> in which it is inscribed.
G.154 inscribe a circle in a triangle
Inscribe a circle in a given triangle.
G.156 circle circumscribed about a polygon
A:term:circle <G.10> is said to be:term:**circumscribed about a figure** <G.155> when the arc of the circle passes through each:term:angle <G.22> of the:term:figure <G.9> about which it is circumscribed.
G.157 circumscribe a circle about a triangle
Circumscribe a circle about 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.161 circumscribe a circle about a square
Circumscribe a circle about 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.163 inscribe a regular pentagon in a circle
Inscribe an equilateral and equiangular pentagon in a given circle.
G.164 circumscribe a regular pentagon about a circle
Circumscribe an equilateral and equiangular pentagon about 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.168 inscribe a regular pentadecagon in a circle
Inscribe an equilateral and equiangular fifteen-angled figure in a given circle.
G.223 mean proportional
To two given segments to find a mean proportional.
G.246 equal circles angles
In equal circles angles have the same ratio as the arcs on which they stand, whether they stand at the centres or at the arcs.
G.385 magnitudes commensurable incommensurable
If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining one is incommensurable with the same.
G.403 find two rational segments commensurable in square only
Find two rational segments commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a segment commensurable in length with the greater.
G.404 find two rational segments commensurable in square only
Find two rational segments commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a segment incommensurable in length with the greater.
G.407 find two segments incommensurable in square which make
Find two segments incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.
G.408 find two segments incommensurable in square which make
Find two segments incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.
G.409 find two segments incommensurable in square which make
Find two segments incommensurable in square which make the sum of the squares on them medial and the rectangle contained by them medial and moreover incommensurable with the sum of the squares on them.
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.509 segments parallel points
If two segments be parallel and points be taken at random on each of them, the segment joining the points is in the same plane with the parallel lines.
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.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.563 cone part cylinder
Any cone is a third part of the cylinder which has the same base with it and equal height.
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.566 base of a cone
And the:term:**base** <base-of-cone> is the:term:circle <G.10> described by the:term:segment <G.4> which is carried round.
G.568 bases of a cylinder
And the:term:**bases** <bases-of-cylinder> are the:term:circles <G.10> described by the two sides opposite to one another which are carried round.
G.570 similar cones cylinders
Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.
G.571 cylinder cut plane
If a cylinder be cut by a plane which is parallel to its opposite planes, then, as the cylinder is to the cylinder, so will the axis be to the axis.
G.572 cones cylinders equal
Cones and cylinders which are on equal bases are to one another as their heights.
G.573 equal cones cylinders
In equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal.
G.574 circles center inscribe
Given two circles about the same center, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.
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.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.
G.587 side hexagon decagon
If the side of the hexagon and that of the decagon inscribed in the same circle be added together, the whole segment has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.
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.
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.590 equilateral triangle inscribed
If an :term:`equilateral triangle <G.13>` be :term:`inscribed in a circle <G.148>`, the :term:`square <G.63>` on the side of the :term:`triangle <G.12>` is triple of the square on the radius of the :term:`circle <G.10>`.
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.601 set out sides
To set out the sides of the five figures and to compare them with one another.