"""F"" can also be derived, which will solve the problem. Its vertices include the ""n"" points on the circle, referred to as the exterior vertices, as well as the interior vertices, the intersections of distinct chords in the interior of the circle. The ""generic intersection"" assumption made above guarantees that each interior vertex is the intersection of no more than two chords. Thus the main task in determining ""V"" is finding the number of interior vertices. As a consequence of the lemma, any two intersecting chords will uniquely determine an interior vertex. These chords are in turn uniquely determined by"
"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,"
"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"
"Chord (geometry) A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just ""secant"", is the infinite line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse. A chord that passes through a circle's center point is the circle's diameter. Every diameter is a chord, but not every chord is a diameter. The word ""chord"" is from the Latin ""chorda"" meaning ""bowstring"". Among properties of chords of a circle are the following: The midpoints of a"
"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"
"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"
"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"
"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"
"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"""
"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"
"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"
"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"
"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"
"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"
"consisting of the joins of corresponding points of two related ranges (points on a line) on different bases (the lines the points are on). Such an envelope is called a line conic (or dual conic). In the real projective plane, a point conic has the property that every line meets it in two points (which may coincide, or may be complex) and any set of points with this property is a point conic. It follows dually that a line conic has two of its lines through every point and any envelope of lines with this property is a line conic."
"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"
"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"
"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"
"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"
"Diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ""diagonal"" derives from the ancient Greek διαγώνιος ""diagonios"", ""from angle to angle"" (from διά- ""dia-"", ""through"", ""across"" and γωνία ""gonia"", ""angle"", related to ""gony"" ""knee""); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as ""diagonus"" (""slanting line""). In matrix algebra, a diagonal of a square matrix is"
"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"
"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"
"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 the hypercycle are called the ""normals"" of"
"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"
"Cross A cross is a geometrical figure consisting of two intersecting lines or bars, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is also termed a saltire in heraldic terminology. The word ""cross"" is recorded in 10th-century Old English as ""cros"", exclusively for the instrument of Christ's crucifixion, replacing the native Old English word ""rood"". The word's history is complicated; it appears to have entered English from Old Irish, possibly via Old Norse, ultimately from the Latin ""crux"" (or its accusative ""crucem"" and"
"Contour line A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional graph of the function ""f""(""x"", ""y"") parallel to the ""x"", ""y"" plane. In cartography, a contour line (often just called a ""contour"") joins points of equal elevation (height) above a given level, such as mean sea level. A contour map is a map illustrated with contour lines, for example a topographic map, which thus shows valleys"
"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"
"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"
"exactly one line. This concept can be generalized to systems of points and circles, with each circle being determined by three non-collinear points. However, three collinear points determine a line, not a circle. This drawback can be removed by adding a point at infinity to every line. If we call both circles and such completed lines ""cycles"", we get an incidence structure in which every three points determine exactly one cycle. In an affine plane the parallel relation between lines is essential. In the geometry of cycles, this relation is generalized to the ""touching"" relation. Two cycles ""touch"" each other"
"Great circle A great circle, also known as an orthodrome, of a sphere is the intersection of the sphere and a plane that passes through the center point of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a ""small circle"", that is, the intersection of the sphere and a plane that"
"of that name. This geometric object can also be described as the set of all points swept by a line that intercepts the axis and rotates around it; or the union of all lines that intersect the axis at a fixed point formula_3 and at a fixed angle formula_4. The ""aperture"" of the cone is the angle formula_5. More generally, when the directrix formula_1 is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of formula_1, one obtains a ""conical quadric"", which is a special case of a quadric surface. A cylindrical"
"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."
"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"