What is the straight line called which joins two points on the circumference of a circle but does not pass not through the centre?
"Secant line In geometry, a secant of a curve is a line that intersects the curve in at least two (distinct) points. The word ""secant"" comes from the Latin word ""secare"", meaning ""to cut"". In the case of a circle, a secant will intersect the circle in exactly two points and a chord is the line segment determined by these two points, that is the interval on a secant whose endpoints are these points. A straight line can intersect a circle in two, one or zero points. A line intersecting in two points is called a ""secant line"", in one"
"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"
"there is exactly one circle that can be drawn through three given points. The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles. To show that a non-trivial intersection of two spheres is a circle, assume (without loss of generality) that one sphere (with radius formula_1) is centered at the origin. Points on this sphere satisfy Also without loss of generality, assume that the second sphere, with radius formula_3, is centered at a point on the positive x-axis, at distance formula_4 from the origin. Its points"
"intersection of the vertical line and the ""x""-axis which passes through the given central point. Draw a line tangent to the circle which passes through the given non-central point. Draw a horizontal line through that point of tangency and find its intersection with the vertical line. The midpoint between that intersection and the given non-central point is the center of the model circle. Draw the model circle around that new center and passing through the given non-central point. Drop a perpendicular ""p"" from the Euclidean center of the circle to the ""x""-axis. Let point ""q"" be the intersection of this"
"fact the only exception) to the dual generation theorem stated above. This degenerate case, in which the curve is a straight-line segment, underlies the Tusi-couple. Centered trochoid In geometry, a centered trochoid is the roulette formed by a circle rolling along another circle. That is, it is the path traced by a point attached to a circle as the circle rolls without slipping along a fixed circle. The term encompasses both epitrochoid and hypotrochoid. The center of this curve is defined to be the center of the fixed circle. Alternatively, a centered trochoid can be defined as the path traced"
"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"
"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"
"does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere. For most pairs of points on the surface of a sphere, there is a unique great circle through the two points. The exception is a pair of antipodal points, for which there are infinitely many great circles. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense, the minor arc is analogous to “straight lines” in Euclidean geometry. The length of the minor arc of a great circle is taken as the"
"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"
"two circles in the Two points, one line problem where the line through P and Q is not parallel to the given line l, can be constructed with compass and straightedge by: CPP problems generally have 2 solutions. Consider a circle centered on one given point P that passes through the second point, Q. Since the solution circle must pass through P, inversion in this circle transforms the solution circle into a line lambda. The same inversion transforms Q into itself, and (in general) the given circle ""C"" into another circle ""c"". Thus, the problem becomes that of finding a"
"Centered trochoid In geometry, a centered trochoid is the roulette formed by a circle rolling along another circle. That is, it is the path traced by a point attached to a circle as the circle rolls without slipping along a fixed circle. The term encompasses both epitrochoid and hypotrochoid. The center of this curve is defined to be the center of the fixed circle. Alternatively, a centered trochoid can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle. Specifically, a centered trochoid is a curve that can be"
"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"
"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"
"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,"
"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"
"Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. This"
"lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map projections. In technical language, these transformations do not change the incidence structure of the tangent line and circle, even though the line and circle may be deformed. The radius of a circle is perpendicular to the tangent line through its endpoint on the circle's circumference. Conversely, the perpendicular to a radius through the same endpoint is a tangent line. The resulting geometrical figure of"
"by three points formula_10 not on a line is obtained by a conversion of the ""3-point-form of a circle's equation"" Homogeneous form<br> In homogeneous coordinates, each conic section with the equation of a circle has the form It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points ""I""(1: ""i"": 0) and ""J""(1: −""i"": 0). These points are called the circular points at infinity. In polar coordinates, the equation of a circle is: where ""a"" is the radius of the circle, formula_14 is the polar coordinate of"
"that a secant line may cross the curve at the two points it intersects it. One can also consider bitangents that are not lines; for instance, the symmetry set of a curve is the locus of centers of circles that are tangent to the curve in two points. Bitangents to pairs of circles figure prominently in Jakob Steiner's 1826 construction of the Malfatti circles, in the belt problem of calculating the length of a belt connecting two pulleys, in Casey's theorem characterizing sets of four circles with a common tangent circle, and in Monge's theorem on the collinearity of intersection"
"in Lie sphere geometry. Tangent lines to circles In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles. A tangent line ""t"" to a circle ""C"" intersects the circle at a single point"
"red circle with an angular line crossing it. Either overtaking is prohibited for all vehicles or certain kinds of vehicles only (e.g. lorries, motorcycles, etc...). In the USA, this is usually phrased as ""no passing zone"" and indicated by a rectangular, black-on-white sign on the right side of the road that says ""DO NOT PASS"" and/or by a solid yellow line painted on the roadway marking the left limit of traffic (centerline), and sometimes supplemented by a yellow (no passing zone) sign on the opposite side of the road (where it can be seen by a driver who is attempting"
"The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles. The orthocenter (blue point), center of the nine-point circle (red), centroid (orange), and circumcenter (green) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. The center of the incircle is not in general located on Euler's line. If one"
"has the Euler line as its directrix. Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. Euler showed in 1765 that in any triangle, the"
"the circle that passes through the three points P, Q and T. If the line ""m"" is ""not"" parallel to the given line ""l"", then it intersects ""l"" at a point G. By the power of a point theorem, the distance from G to a tangent point T must equal the geometric mean Two points on the given line ""L"" are located at a distance formula_12 from the point G, which may be denoted as T and T. The two solution circles are the circles that pass through the three points (P, Q, T) and (P, Q, T), respectively. The"
"one solution cannot have a 2 in a corner with two non-diagonally adjacent, unconstrained squares, and a diagonally adjacent 0 or 1.) In the figure below, the circled points can be connected by a line directly between them, and also by a line that traverses the other three sides of the square that extends to the left of the points. It should be clear (with the red line ignored) that for both paths the remainder of the solution can be the same – since the constraints for the remainder of the solution are the same – so both paths are"
"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"
"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"
"Cross section (geometry) In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross sections. The boundary of a cross section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is a sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional"
"circular. Alternatively, this can be seen from considering the link diagram: if one assumes that circles 1 and 2 touch at their two crossing points, then they either lie in a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible; see . It is, however, true that one can use ellipses (right picture). These may be taken to be of arbitrarily small eccentricity; i.e. no matter how close to being circular their shape may be, as long as they are not perfectly circular,"
"= ½ the Miquel point is the circumcentre . The theorem can be reversed to say: for three circles intersecting at ""M"", a line can be drawn from any point ""A"" on one circle, through its intersection ""C´"" with another to give ""B"" (at the second intersection). ""B"" is then similarly connected, via intersection at ""A´"" of the second and third circles, giving point ""C"". Points ""C"", ""A"" and the remaining point of intersection, ""B´"", will then be collinear, and triangle ""ABC"" will always pass through the circle intersections ""A´"", ""B´"" and ""C´"". If the inscribed triangle ""XYZ"" is similar"
"Circumgon In mathematics and particularly in elementary geometry, a circumgon is a geometric figure which circumscribes some circle, in the sense that it is the union of the outer edges of non-overlapping triangles each of which has a vertex at the center of the circle and opposite side on a line that is tangent to the circle. The limiting case in which part or all of the circumgon is a circular arc is permitted. A circumgonal region is the union of those triangular regions. Every triangle is a circumgonal region because it circumscribes the circle known as the incircle of"
"one. Creatures that are not two-legged and cannot fly (for example, whales and spiders) would all be represented by points outside both circles. The combined region of sets A and B is called the ""union"" of A and B, denoted by . The union in this case contains all living creatures that are either two-legged or that can fly (or both). The region in both A and B, where the two sets overlap, is called the ""intersection"" of A and B, denoted by . For example, the intersection of the two sets is not empty, because there ""are"" points that"
"Tangent lines to circles In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles. A tangent line ""t"" to a circle ""C"" intersects the circle at a single point T. For comparison, secant"