"point a ""tangent line"" and in no points an ""exterior line"". A ""chord"" of a circle is the line segment that joins two distinct points of the circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in his treatment, are usually proved. For instance, ""Theorem (Elementary Circular Continuity)"": If formula_1 is a circle and formula_2 a line that contains a point that is inside formula_1 and a point that is outside of"
"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"
"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"
"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"
"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."
"a secant line of the circle, and the line segment connecting the intersection points is called a chord of the circle. If formula_33 holds, there exists only one intersection point and the line is tangent to the circle. If the weak inequality does not hold, the line does not intersect the circle. If the circle's midpoint is not the origin, see. The intersection of a line and a parabola or hyperbola may be treated analogously. The determination of the intersection points of two circles can be reduced to the previous case of intersecting a line and a circle. By subtraction"
"Limiting point (geometry) In geometry, the limiting points of two disjoint circles ""A"" and ""B"" in the Euclidean plane are points ""p"" that may be defined by any of the following equivalent properties: The midpoint of the two limiting points is the point where the radical axis of ""A"" and ""B"" crosses the line through their centers. This intersection point has equal power distance to all the circles in the pencil containing ""A"" and ""B"". The limiting points themselves can be found at this distance on either side of the intersection point, on the line through the two circle centers."
"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"
"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"
"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"
"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"
"Line segment In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent"
"Newton line In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides. The line segments ""GH"" and ""IJ"" that connect the midpoints of opposite sides (the bimedians) of a convex quadrilateral intersect in a point that lies on the Newton line. This point ""K"" bisects the line segment ""EF"" that connects the diagonal midpoints. By Anne's theorem and its converse, any interior point ""P"" on the Newton line of a quadrilateral ""ABCD"" has the property that where [""ABP""] denote the area of triangle"
"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"
"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"
"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"
"as a subset ""L"" that can be parametrized as for some vectors formula_4. Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points. In geometry, it is sometimes defined that a point ""B"" is between two other points ""A"" and ""C"", if the distance ""AB"" added to the distance ""BC"" is equal to the distance ""AC"". Thus in formula_7 the line segment with endpoints and is the following collection of points: In an axiomatic treatment of geometry, the notion of betweenness is"
"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"
"i.e., they are cyclical rather than linear. This is true of the line at infinity itself; it meets itself at its two endpoints (which are therefore not actually endpoints at all) and so it is actually cyclical. The line at infinity can be visualized as a circle which surrounds the affine plane. However, diametrically opposite points of the circle are equivalent—they are the same point. The combination of the affine plane and the line at infinity makes the real projective plane, formula_1. A hyperbola can be seen as a closed curve which intersects the line at infinity in two different"
"the three given circles until they just touch, i.e., are tangent. An inversion in their tangent point with respect to a circle of appropriate radius transforms the two touching given circles into two parallel lines, and the third given circle into another circle. Thus, the solutions may be found by sliding a circle of constant radius between two parallel lines until it contacts the transformed third circle. Re-inversion produces the corresponding solutions to the original problem. The concept of a tangent line to one or more circles can be generalized in several ways. First, the conjugate relationship between tangent points"
"A real plane contains these two points if and only if it is perpendicular to the axis of revolution. Thus the circular sections are the plane sections by a plane perpendicular to the axis, that have at least two real points. In the other cases, the intersection of with the ombilic consists of two different pairs of complex conjugate points. As is a curve of degree two, its intersection with the plane at infinity consists of two points, possibly equal. The curve is thus a circle, if these two points are one of these two pairs of complex conjugate points"
"a line through any two of its points. Such a point can be found by drawing a circle C that intersects both circles A and B in two points. The two lines passing through each pair of intersection points are the radical axes of A and C and of B and C. These two lines intersect in a point J that is the radical center of all three circles, as described above; therefore, this point also lies on the radical axis of A and B. Repeating this process with another such circle D provides a second point K. The radical"
"Line–line intersection In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line. Distinguishing these cases and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection. In three-dimensional Euclidean geometry, if two lines are not in the same plane they are called skew lines and have no point of intersection. If they are in the same plane there are three possibilities: if they coincide (are not distinct lines) they have an infinitude of points in common (namely all of the points on either"
"Edge (geometry) In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment on the boundary, and is often called a side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal. In graph theory, an edge is an abstract object connecting two graph vertices, unlike polygon and polyhedron edges which"
"intersection point P of the two lines (2). The intersection points of this circle with the two given lines (5) are T1 and T2. Two circles of the same radius, centered on T1 and T2, intersect at points P and Q. The line through P and Q (1) is an angle bisector. Rays have one angle bisector; lines have two, perpendicular to one another. A few basic results are helpful in solving special cases of Apollonius' problem. Note that a line and a point can be thought of as circles of infinitely large and infinitely small radius, respectively. PPP problems"
"be done by choosing one of the three intersection points P between the given lines; and drawing a circle centered on the midpoint of C and P of diameter equal to the distance between C and P. The intersections of that circle with the intersecting given lines are the two points of tangency. PLL problems generally have 2 solutions. As shown above, if a circle is tangent to two given lines, its center must lie on one of the two lines that bisect the angle between the two given lines. By symmetry, if such a circle passes through a given"
"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"
"Twin circles In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of , , and , perpendicular to line , then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment. These circles first appeared"
"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 unit circle in the complex plane. If ""C"" is the midpoint of the segment ""AB"", then the collection of points ""P"" satisfying the Apollonius condition is not a circle, but rather a line. Thus, if ""A"", ""B"", and ""C"" are given distinct points in the plane, then the locus of points ""P"" satisfying the above equation is called a ""generalised circle."" It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius. In every triangle a unique circle, called the incircle, can be inscribed such that it"
"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"
"the circle is mapped to three-dimensional Euclidean space by an injective function (a continuous function that does not map two different points of the circle to the same point of space), its image is a closed curve. Two disjoint closed curves that both lie on the same plane are unlinked, and more generally a pair of disjoint closed curves is said to be unlinked when there is a continuous deformation of space that moves them both onto the same plane, without either curve passing through the other or through itself. If there is no such continuous motion, the two curves"