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Geometry theorem relating line segments created by intersecting secants of a circle
geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated
Intersecting_secants_theorem
Geometry theorem relating the line segments created by intersecting chords in a circle
and point S. Next to the tangent-secant theorem and the intersecting secants theorem, the intersecting chords theorem represents one of the three basic
Intersecting_chords_theorem
Geometry theorem relating line segments created by a secant and tangent line
tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the
Tangent–secant_theorem
Relative distance of a point from a circle
. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: Intersecting secants theorem: For a point
Power_of_a_point
Line that intersects a curve at least twice
Christopher Clavius demonstrated this result, sometimes called the intersecting secants theorem, in their commentaries on Euclid. For curves more complicated
Secant_line
Relates the length of a median of a triangle to the lengths of its sides
In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that the sum of the
Apollonius's_theorem
Simple curve of Euclidean geometry
1/2arc(AQ). The chord theorem states that if two chords, CD and EB, intersect at A, then AC × AD = AB × AE. If two secants, AE and AD, also cut the
Circle
Ancient Greek mathematician (fl. 300 BC)
the later tradition of Alexandria. In the Elements, Euclid deduced the theorems from a small set of axioms. He also wrote works on perspective, conic sections
Euclid
angle theorem (geometry) Intercept theorem (Euclidean geometry) Intersecting chords theorem (Euclidean geometry) Intersecting secants theorem (Euclidean
List_of_theorems
Trigonometric function defined as secant minus one
be found in Book 3 of Euclid's Elements, as used e.g. in the intersecting secants theorem. 18th century sources in Latin called any non-tangential line
Exsecant
Book by Thomas Little Heath
theorem Intersecting chords theorem Intersecting secants theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem
A History of Greek Mathematics
A_History_of_Greek_Mathematics
Problem of constructing equal-area shapes
proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( π {\displaystyle \pi } ) is a transcendental number
Squaring_the_circle
Mathematics of Ancient Greece and the Mediterranean, 5th BC to 6th AD
Greek mathematics is obscure, and traditional narratives of mathematical theorems found before the fifth century BC are regarded as later inventions. It
Ancient_Greek_mathematics
Functions of an angle
tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less commonly used. Each of these
Trigonometric_functions
Ancient Greek mathematician
theorem Intersecting chords theorem Intersecting secants theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem
Leon_(mathematician)
Method of assigning coordinates to every line in projective 3-space
because a, b are neither zero nor parallel (the planes being distinct and intersecting). If point x satisfies both plane equations, then it also satisfies the
Plücker_coordinates
Generalization of Pythagorean theorem
cosines (also known as the cosine formula or cosine rule or Al-Kashi’s theorem) relates the lengths of the sides of a triangle to the cosine of one of
Law_of_cosines
In mathematical theory of differential equations the Chaplygin Theorem states about the existence and uniqueness of the solution to an initial value problem
Chaplygin's Theorem and Method for Solving ODE
Chaplygin's_Theorem_and_Method_for_Solving_ODE
Distance function defined between probability distributions
{\displaystyle f} cannot increase with slope larger than 1. Thus all its secants have slope | f ( x ) − f ( y ) x − y | ≤ 1 {\displaystyle {\bigg |}{\frac
Wasserstein_metric
Geometric line segment whose endpoints lie on a circular arc
extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP · PB = CP · PD (power of a point theorem). The midpoints
Chord_(geometry)
Invariant cord in one of two intersecting circles based on any point in the other
chord theorem 1925 in the article sur deux cercles secants for the Belgian math journal Mathesis. Eight years later he published On Two Intersecting Spheres
Constant_chord_theorem
Ancient Greek spherical geometry treatise
astronomy as modeled by the celestial sphere. Primarily consisting of theorems which were known at least informally a couple centuries earlier, the Spherics
Theodosius'_Spherics
Characterizes spherical triangles with fixed base and area
In spherical geometry, Lexell's theorem holds that every spherical triangle with the same surface area on a fixed base has its apex on a small circle
Lexell's_theorem
Line which touches a circle at exactly one point
equal (this is sometimes called the Two Tangents Theorem, see Incircle). By the secant-tangent theorem, the square of this tangent length equals the power
Tangent_lines_to_circles
Plane curve: conic section
to the intersecting plane, the intersection curve will be a hyperbola (or degenerate hyperbola, if the two generatrices are in the intersecting plane)
Parabola
grid of transversal lines made with secants between two groups of arcs that form two graduated limbs. The secants are drawn by joining the division of
Transversal (instrument making)
Transversal_(instrument_making)
Plane curve
line g {\displaystyle g} intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line, tangent and secant. Through any point of an
Ellipse
Algebraic curve in mathematics
method of tangents and secants detailed above, starting with a finite number of rational points. More precisely the Mordell–Weil theorem states that the group
Elliptic_curve
Mathematical treatise by Euclid
concern intersecting chords and tangents; proposition 35 is the intersecting chords theorem, and proposition 36 is the tangent–secant theorem. Book IV
Euclid's_Elements
clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides
History_of_trigonometry
Area bounded by a circular arc and a straight line
the rest of the disk by a straight line. The complete line is known as a secant, and the section inside the disk as a chord. More formally, a circular segment
Circular_segment
Mathematical term
the secant intersecting y = x2 at (0,0) and (3,9) is 3. (The slope of the tangent at x = 3⁄2 is also 3 − a consequence of the mean value theorem.) By
Slope
In mathematics, straight line touching a plane curve without crossing it
example, for a line passing through the vertex of a triangle and not intersecting it otherwise—where the tangent line does not exist for the reasons explained
Tangent
Mathematical treatise by Archimedes
The Method of Mechanical Theorems (Greek: Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as The Method, is one of the major surviving
The Method of Mechanical Theorems
The_Method_of_Mechanical_Theorems
Unique point and line of a conic section
additional three diagonal points. Given a point Z not on conic C, draw two secants from Z through C crossing at points A, B, D, and E. Then these four points
Pole_and_polar
Part of a circle between two points
segment for details. Using the intersecting chords theorem (also known as power of a point or secant tangent theorem) it is possible to calculate the
Circular_arc
Circle with radius of one
right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 = 1. {\textstyle x^{2}+y^{2}=1
Unit_circle
Geometrical concept
the object do not subtract away, as would be required by the Divergence Theorem applied to the constant vector field r ^ {\displaystyle \mathbf {\hat {r}}
Cross_section_(geometry)
All points whose relative distances to two circles are same
is the common tangent line. The radical axis of two intersecting circles is their common secant line. The radical axis of two touching circles is their
Radical_axis
Type of geometry
harmonic conjugates of P on a variable secant line passing through P and C. Descriptive geometry Fundamental theorem of projective geometry Grassmann–Cayley
Projective_geometry
Concept in projective geometry
(thus, every point not on the arc is on a secant line of the arc–a line meeting the arc in two points.) Theorem: Let K be a complete k-arc in Π = PG(2,q)
Blocking_set
Algebraic variety in a projective space
Riemann–Roch theorem to higher dimension is the Hirzebruch–Riemann–Roch theorem, as well as the far-reaching Grothendieck–Riemann–Roch theorem. Hilbert schemes
Projective_variety
Area of geometry, about angles and lengths
cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: c 2 = a 2 + b 2 − 2 a b cos C , {\displaystyle
Trigonometry
integral . related rates . removable discontinuity . Rolle's theorem . root test . scalar . secant line . second-degree polynomial . second derivative . second
Glossary_of_calculus
Line through four points of a curve
two points; and a trisecant, a line that intersects a curve or surface in three points. Compared to secants and trisecants, quadrisecants are especially
Quadrisecant
and contains "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians
Timeline_of_mathematics
Straight figure with zero width and depth
being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do not intersect or are collinear
Line_(geometry)
Notable events in the history of geometry
classification of cubic equations with geometric solutions found by means of intersecting conic sections." He became the first to find general geometric solutions
Timeline_of_geometry
Study of angle-preserving transformations
orthogonal, then a straight line passing through the center O of k and intersecting q, does so at inverse points with respect to k. Given a triangle OAB
Inversive_geometry
Greek parchment codex manuscript
thought to have been lost (the Ostomachion and the Method of Mechanical Theorems) and the only surviving original Greek edition of his work On Floating
Archimedes_Palimpsest
Polynomial function of degree 4
the substitution y = x2 that two quadratics intersect in four points is an instance of Bézout's theorem. Explicitly, the four points are Pi ≔ (xi, xi2)
Quartic_function
Unsolved problem about inscribing a square in a Jordan curve
with the curves generated in the same way for a perpendicular family of secants, there are an odd number of crossings. Therefore, there always exists at
Inscribed_square_problem
Concept in projective geometry
these are: Desargues' theorem ⇔ Converse of Desargues' theorem Pascal's theorem ⇔ Brianchon's theorem Menelaus' theorem ⇔ Ceva's theorem Not only statements
Duality_(projective_geometry)
Mathematics used in Ancient China
Suanjing contains an in-depth proof of the Gougu Theorem (a special case of the Pythagorean theorem), but focuses more on astronomical calculations. However
Chinese_mathematics
Fundamental trigonometric functions
{\displaystyle \cos(\gamma )=0} , the resulting equation becomes the Pythagorean theorem. The cross product and dot product are operations on two vectors in Euclidean
Sine_and_cosine
Mathematical model combining space and time
because the inverse of the slope—representing the necessary speed—for all secants is less than c {\displaystyle c} . On the other hand, the green hyperbolae
Spacetime
Sphere-like surface
ovoid there is a suitable hyperplane ε {\displaystyle \varepsilon } not intersecting it, one can call this hyperplane the hyperplane ε ∞ {\displaystyle \varepsilon
Ovoid_(projective_geometry)
Point from which two similar geometric figures can be scaled to each other
theorem)}}\end{aligned}}} Segment RQ' is seen in the same angle from P and S', which means R, P, S', Q' lie on a circle. Then from the intersecting chords
Homothetic_center
Surface able to be flattened without distortion
Developable surfaces can be generated in infinite varieties with some general theorems. As previously stated, the tangent developable surfaces are constructed
Developable_surface
Algorithm for finding zeros of functions
Kantorovich theorem Laguerre's method Methods of computing square roots Newton's method in optimization Richardson extrapolation Root-finding algorithm Secant method
Newton's_method
Locus of the zeros of a polynomial of degree two
fact two complex conjugate intersecting planes). For ε 3 = 0 , {\displaystyle \varepsilon _{3}=0,} one has two intersecting planes (reducible quadric)
Quadric
Greek mathematician and physicist (c. 287 – 212 BC)
the method of exhaustion to derive and rigorously prove many geometrical theorems, including the area of a circle, the surface area and volume of a sphere
Archimedes
projective space meeting a variety in n+1 points. 2. A secant variety is the union of the secants of a variety. second kind All residues at poles are zero
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Line tangent to a curve at two locations
algebraic curve will have infinitely many secant lines, but only finitely many bitangents. Bézout's theorem implies that an algebraic plane curve with
Bitangent
Construction of an angle equal to one third a given angle
reducible over by Q then it has a rational root. By the rational root theorem, this root must be ±1, ±1/2, ±1/4 or ±1/8, but none of these is a
Angle_trisection
Mathematical instrument consisting of two hinged rulers
triangle is A t = c ( r − h ) {\displaystyle A_{t}=c(r-h)} . Using Pythogras' theorem, we can show that r = ( c 2 + h 2 ) / 2 h {\displaystyle r=(c^{2}+h^{2})/2h}
Sector_(instrument)
It can be derived from the mean value theorem by considering the secant of the graph of the function x ↦ x ⋅ ln x {\displaystyle
Identric_mean
Mathematical set with some added structure
succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups. Since that time, new theorems of classical geometry
Space_(mathematics)
Curve traced by a point on a rod as one end is dragged along a line
if the coordinates of the object are (x, y), then by the Pythagorean theorem the y-coordinate of the puller is y + a 2 − x 2 {\displaystyle y+{\sqrt
Tractrix
theoretical circle-packing given by the Koebe-Andreev-Thurston theorem). See also Fáry's theorem on straight-line drawings of planar graphs. Force-based algorithms
List_of_algorithms
Isometric automorphisms of a hyperbolic space
the length of the triangle hypotenuse is sec a, where sec denotes the secant function. Set r = sec a and apply the third fundamental hyperbolic motion
Hyperbolic_motion
Change of variable for integrals involving trigonometric functions
{x}{2}}\\[6pt]&={\frac {2t^{2}}{2t}}=t\\[6pt]&=\tan {\tfrac {x}{2}}\end{aligned}}} The secant integral may be evaluated in a similar manner. We wish to evaluate the integral:
Tangent half-angle substitution
Tangent_half-angle_substitution
Algorithms for calculating square roots
may be used as the approximation, but a least-squares regression line intersecting the arc will be more accurate. A least-squares regression line minimizes
Square_root_algorithms
expansion, respectively. Norton's theorem In direct-current circuit theory, Norton's theorem (aka Mayer–Norton theorem) is a simplification that can be
Glossary_of_engineering:_M–Z
Pappus's hexagon theorem holds. The following result, due to Francis Buekenhout, is an astonishing statement for finite projective spaces. Theorem: Let be P
Quadratic_set
as 1961. A simpler proof was later given by Louis J. Mordell. Euler's theorem in geometry List of triangle inequalities Erdős, Paul; Mordell, L. J.;
Barrow's_inequality
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INTERSECTING SECANTS-THEOREM
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INTERSECTING SECANTS-THEOREM
INTERSECTING SECANTS-THEOREM
INTERSECTING SECANTS-THEOREM
INTERSECTING SECANTS-THEOREM
INTERSECTING SECANTS-THEOREM
INTERSECTING SECANTS-THEOREM
INTERSECTING SECANTS-THEOREM
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