"P intersects the circle at A and B, the endpoints of the diameter. Their respective products of distances are equal to each other and to the power of point P in that circle This equality is sometimes known as the """"secant-tangent theorem"""", """"intersecting chords theorem"""", or the """"power-of-a-point theorem"""". The power of a point is used in many geometrical definitions and proofs. For example, the radical axis of two given circles is the straight line consisting of points that have equal power to both circles. For each point on this line, there is a unique circle centered on that point"
"Circle A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior"
"line cannot be drawn without a straightedge. A line is considered to be given by any two points, as any two points define a line uniquely, and a unique line can be defined by any two points on it. This can be done with compass alone quite naturally. There is nothing to prove. This construction can be done directly with a compass provided the centers and radii of the two circles are known. Due to the compass-only construction of the center of a circle (given below), it can always be assumed that any circle is described by its center and"
"Generalised circle A generalized circle, also referred to as a ""cline"" or ""circline"", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and are best treated together. Inversive plane geometry is formulated on the plane extended by one point at infinity. A straight line is then thought of as one of the circles that passes through the asymptotic point at infinity. The fundamental transformations in inversive geometry, the ""inversions"", have the property that they map generalized circles to generalized circles. Möbius transformations,"
"on the point on the circle and rotate the compass. Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ""ratio"" (other than 1) of distances to two fixed foci, ""A"" and ""B"". (The set of points where the distances are equal is the perpendicular bisector of segment ""AB"", a line.) That circle is sometimes said to be drawn ""about"" two points. The proof is in two parts. First, one must prove that, given two foci ""A"" and ""B"" and a ratio of distances, any point ""P"" satisfying"
"Radical axis The radical axis (or power line) of two circles is the locus of points at which tangents drawn to both circles have the same length. The radical axis is always a straight line and always perpendicular to the line connecting the centers of the circles, albeit closer to the circumference of the larger circle. If the circles intersect, the radical axis is the line passing through the intersection points; similarly, if the circles are tangent, the radical axis is simply the common tangent. For any point P on the radical axis, there is a unique circle centered on"
"Circumference In geometry, the circumference (from Latin ""circumferentia"", meaning ""carrying around"") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses. Informally, ""circumference"" may also refer to the edge itself rather than to the"
"More generally, there is a ""k""-dimensional tangent space at each point of a ""k""-dimensional manifold in the ""n""-dimensional Euclidean space. Tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that ""just touches"" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where ""f"""
"Tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that ""just touches"" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where ""f"" is the derivative of ""f"". A similar definition applies to space curves and curves in ""n""-dimensional Euclidean space. As it"
"Line (geometry) The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined as the ""[…] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its"
"of two lines. The non-degenerate circles are called ""smooth circles"", while the degenerate ones are called ""singular circles."" There are two types of singular circles. One is the union of the line at infinity with another line in the projective plane (possibly the line at infinity again), and the other is union of two lines in the projective plane, one through each of the two circular points. These are the limits of smooth circles as the radius tends to and , respectively. In the latter case, no point on either of the two lines has real coordinates except for the"
"Great-circle distance The great-circle distance or orthodromic distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called ""great circles"". The determination of"
"circle and tangent line has a reflection symmetry about the axis of the radius. No tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, ""two"" tangent lines can be drawn to a circle from a point P outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining P to the center point O of the circle. Thus the lengths of the segments from P to the two tangent points are equal. By the secant-tangent"
"Director circle In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other. The director circle of an ellipse circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius formula_1, where formula_2 and formula_3 are the semi-major axis and semi-minor axis of the ellipse. Additionally, it has the property that, when viewed from any point on the circle, the ellipse spans a right angle. The"
"Trochoid A trochoid (from the Greek word for wheel, ""trochos"") is the curve described by a fixed point on a circle as it rolls along a straight line. The cycloid is a notable member of the trochoid family. The word ""trochoid"" was coined by Gilles de Roberval. As a circle of radius ""a"" rolls without slipping along a line ""L,"" the center ""C"" moves parallel to ""L,"" and every other point ""P"" in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let ""CP = b"". Parametric equations of the trochoid for which ""L"" is"
"have where again formula_12 is the length of the semi-major axis and formula_13 is the eccentricity formula_14 In graph theory the circumference of a graph refers to the longest (simple) cycle contained in that graph. Circumference In geometry, the circumference (from Latin ""circumferentia"", meaning ""carrying around"") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length"
"where the line crosses the formula_37 axis. Depending on the context, either formula_42 or the point formula_138 is called the formula_37-intercept. In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that ""just touches"" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where ""f"" is"
"the most commonly encountered constructions are explained. A perpendicular bisector of a side of a triangle is a straight line passing through the midpoint of the side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter, usually denoted by O; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle, called the ""circumdiameter"", can be found from the law of sines stated above. The circumcircle's radius is called the ""circumradius"". Thales' theorem implies that if"
"that places ""p"" onto ""l"" and passes through ""p"". This axiom is equivalent to finding the intersection of a line with a circle, so it may have 0, 1, or 2 solutions. The line is defined by ""l"", and the circle has its center at ""p"", and a radius equal to the distance from ""p"" to ""p"". If the line does not intersect the circle, there are no solutions. If the line is tangent to the circle, there is one solution, and if the line intersects the circle in two places, there are two solutions. If we know two points"
"ellipse. The Apollonius circle, the locus of points ""X"" such that the ratio of distances of ""X"" to two foci ""P"" and ""P"" is a fixed constant ""r"", is another special case, with ""w"" = 1, ""w"" = −""r"", and ""C"" = 0. In the case of a parabola the director circle degenerates to a straight line, the directrix of the parabola. Director circle In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross"
"describes the unit circle. For two geometric objects P and Q represented by the relations formula_69 and formula_70 the intersection is the collection of all points formula_71 which are in both relations. For example, formula_5 might be the circle with radius 1 and center formula_73: formula_74 and formula_75 might be the circle with radius 1 and center formula_76. The intersection of these two circles is the collection of points which make both equations true. Does the point formula_73 make both equations true? Using formula_73 for formula_71, the equation for formula_75 becomes formula_81 or formula_82 which is true, so formula_73 is"
"an arbitrary line in a projective plane that shall be referred to as the absolute line. Select two distinct points on the absolute line and refer to them as absolute points. Several metrical concepts can be defined with reference to these choices. For instance, given a line containing the points and , the midpoint of line segment is defined as the point which is the projective harmonic conjugate of the point of intersection of and the absolute line, with respect to and . A conic in a projective plane that contains the two absolute points is called a circle. Since"
"Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of ""the"" diameter rather than ""a"" diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius"
"the same points, but at right angles. Hypercycle (geometry) In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L, one can construct a hypercycle by taking all points Q on the same side of L as P, with perpendicular distance to L equal to that of P. The line L is called the ""axis"", ""center"", or ""base line"" of the hypercycle. The lines perpendicular to the ""axis"", which is also perpendicular to"
"It ""is"" possible to do them with straightedge alone given one circle and its center. Straightedge A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler. Straightedges are used in the automotive service and machining industry to check the flatness of machined mating surfaces. True straightness can in some cases be checked by using a laser line level as an optical straightedge: it can illuminate an accurately straight line on a flat surface such as the edge of"
"points satisfy the homogeneous equations of the type The case where the coefficients are all real gives the equation of a general circle (of the real projective plane). In general, an algebraic curve that passes through these two points is called circular. The circular points at infinity are the points at infinity of the isotropic lines. They are invariant under translations and rotations of the plane. The concept of angle can be defined using the circular points, natural logarithm and cross-ratio: Sommerville configures two lines on the origin as formula_3 Denoting the circular points as ""ω"" and ω""′, he obtains"
"Osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ""p"" on the curve has been traditionally defined as the circle passing through ""p"" and a pair of additional points on the curve infinitesimally close to ""p"". Its center lies on the inner normal line, and its curvature is the same as that of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named ""circulus osculans"" (Latin for ""kissing circle"") by Leibniz."
"Philo line In geometry, the Philo line is a line segment defined from an angle and a point. The Philo line for a point ""P"" that lies inside an angle with edges ""d"" and ""e"" is the shortest line segment that passes through ""P"" and has its endpoints on ""d"" and ""e"". Also known as the Philon line, it is named after Philo of Byzantium, a Greek writer on mechanical devices, who lived probably during the 1st or 2nd century BC. The Philo line is not, in general, constructible by compass and straightedge. Philo's line can be used to double"
"Squircle A squircle is a mathematical shape intermediate between a square and a circle. It is a special case of a superellipse. The word ""squircle"" is a portmanteau of the words ""square"" and ""circle"". In a Cartesian coordinate system, the squircle centred on the point (""a"", ""b"") with axes parallel to the coordinate axes is described by the equation: where ""r"" is the minor radius of the squircle (cf. equation of a circle). The case that is centred on the origin (that is, with ""a"" = ""b"" = 0) is called Lamé's special quartic. When ""a"" = ""b"" = 0"
"Circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. A polygon that has a circumscribed circle is called a cyclic polygon (sometimes a concyclic polygon, because the vertices are concyclic). All regular simple polygons, all isosceles trapezoids, all triangles and all rectangles are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. Not every"
"Intersection (Euclidean geometry) In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the intersection of two distinct lines, which either is one point or does not exist if the lines are parallel. Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved"
"length of the edge. The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this can not be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms. The circumference of a circle is related to one of the most important mathematical constants. This constant,"
"Concyclic points In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, but four or more such points in the plane are not necessarily concyclic. In general the centre ""O"" of a circle on which points ""P"" and ""Q"" lie must be such that ""OP"" and ""OQ"" are equal distances. Therefore ""O"" must lie on the perpendicular bisector of the line"