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Power Of A Point
In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826. Specifically, the power \Pi(P) of a point P with respect to a circle c with center O and radius r is defined by : \Pi(P)=, PO, ^2 - r^2. If P is ''outside'' the circle, then \Pi(P)>0, if P is ''on'' the circle, then \Pi(P)=0 and if P is ''inside'' the circle, then \Pi(P)<0. Due to the Pythagorean theorem the number has the simple geometric meanings shown in the diagram: For a point outside the circle is the squared tangential distance of point to the circle . Points with equal power, |
Jakob Steiner
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards studied at Heidelberg. Then, he went to Berlin, earning a livelihood there, as in Heidelberg, by tutoring. Here he became acquainted with A. L. Crelle, who, encouraged by his ability and by that of Niels Henrik Abel, then also staying at Berlin, founded his famous ''Journal'' (1826). After Steiner's publication (1832) of his ''Systematische Entwickelungen'' he received, through Carl Gustav Jacob Jacobi, who was then professor at Königsberg University, and earned an honorary degree there; and through the influence of Jacobi and of the brothers Alexander and Wilhelm von Humboldt a new chair of geometry was founded for him at Berlin (1834). This he occupied until his death in Bern on 1 April 1863. He was described by Thomas Hirst as follows: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersecting Chords Theorem
The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's ''Elements''. More precisely, for two chords ''AC'' and ''BD'' intersecting in a point ''S'' the following equation holds: :, AS, \cdot, SC, =, BS, \cdot, SD, The converse is true as well, that is if for two line segments ''AC'' and ''BD'' intersecting in S the equation above holds true, then their four endpoints ''A'', ''B'', ''C'' and ''D'' lie on a common circle. Or in other words if the diagonals of a quadrilateral ''ABCD'' intersect in ''S'' and fulfill the equation above then it is a cyclic quadrilateral. The value of the two products in the chord theorem depends only on the distance of the intersection point ''S'' from the circle's center a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent-secant Theorem
The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's ''Elements''. Given a secant ''g'' intersecting the circle at points G1 and G2 and a tangent ''t'' intersecting the circle at point ''T'' and given that ''g'' and ''t'' intersect at point ''P'', the following equation holds: :, PT, ^2=, PG_1, \cdot, PG_2, The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle. For two lines ''AD'' and ''BC'' that intersect each other in ''P'' and some circle in '' ..., the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersecting Chords Theorem
The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's ''Elements''. More precisely, for two chords ''AC'' and ''BD'' intersecting in a point ''S'' the following equation holds: :, AS, \cdot, SC, =, BS, \cdot, SD, The converse is true as well, that is if for two line segments ''AC'' and ''BD'' intersecting in S the equation above holds true, then their four endpoints ''A'', ''B'', ''C'' and ''D'' lie on a common circle. Or in other words if the diagonals of a quadrilateral ''ABCD'' intersect in ''S'' and fulfill the equation above then it is a cyclic quadrilateral. The value of the two products in the chord theorem depends only on the distance of the intersection point ''S'' from the circle's center a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersecting Secants Theorem
The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle. For two lines ''AD'' and ''BC'' that intersect each other in ''P'' and some circle in ''A'' and ''D'' respective ''B'' and ''C'' the following equation holds: :, PA, \cdot, PD, =, PB, \cdot, PC, The theorem follows directly from the fact, that the triangles PAC and PBD are similar. They share \angle DPC and \angle ADB=\angle ACB as they are inscribed angles over AB. The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: :\frac=\frac \Leftrightarrow , PA, \cdot, PD, =, PB, \cdot, PC, Next to the intersecting chords theorem and the tangent-secant theorem The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's ''Elements''. Giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states :c^2 = a^2 + b^2 - 2ab\cos\gamma, where denotes the angle contained between sides of lengths and and opposite the side of length . For the same figure, the other two relations are analogous: :a^2=b^2+c^2-2bc\cos\alpha, :b^2=a^2+c^2-2ac\cos\beta. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90 degrees, or radians), then , and thus the law of cosines reduces to the Pythagorean theorem: :c^2 = a^2 + b^2. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known. History Though the notion of the cosine was not yet developed in his time, Euclid's '' Elements'', dating back to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kreise2-aehnlpunkte-var
Kreis is the German word for circle. Kreis may also refer to: Places * , or circles, various subdivisions roughly equivalent to counties, districts or municipalities ** Districts of Germany (including and ) ** Former districts of Prussia, also known as ** ''Kreise'' of the former Electorate of Saxony *, or Imperial Circles, ceremonial associations of several regional monarchies () and/or imperial cities () in the Holy Roman Empire People * Harold Kreis (born 1959), Canadian-German ice hockey coach * Jason Kreis (born 1972), American soccer player * Melanie Kreis (born 1971), German businesswoman * Wilhelm Kreis (1873–1955), German architect Music and culture *''Der Kreis'', a Swiss gay magazine * ''Kreise'' (album), a 2017 album by Johannes Oerding See also * Krai, an administrative division in Russia * Kraj, an administrative division in Czechia and Slovakia * Okręg, an administrative division in Poland * Okres, an administrative division in Czechia and Slov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similarity (geometry)
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (geometry), scaling (enlarging or reducing), possibly with additional translation (geometry), translation, rotation (mathematics), rotation and reflection (mathematics), reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruence (geometry), congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. If two angles of a triangle h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homothety
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by the rule : \overrightarrow=k\overrightarrow for a fixed number k\ne 0. Using position vectors: :\mathbf x'=\mathbf s + k(\mathbf x -\mathbf s). In case of S=O (Origin): :\mathbf x'=k\mathbf x, which is a uniform scaling and shows the meaning of special choices for k: :for k=1 one gets the ''identity'' mapping, :for k=-1 one gets the ''reflection'' at the center, For 1/k one gets the ''inverse'' mapping defined by k. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if k>0) or reverse (if k<0) the direction of all vectors. Together with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vieta's Theorem
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta"). Basic formulas Any general polynomial of degree ''n'' :P(x) = a_nx^n + a_x^ + \cdots + a_1 x + a_0 (with the coefficients being real or complex numbers and ) has (not necessarily distinct) complex roots by the fundamental theorem of algebra. Vieta's formulas relate the polynomial's coefficients to signed sums of products of the roots as follows: :\begin r_1 + r_2 + \dots + r_ + r_n = -\dfrac \\ (r_1 r_2 + r_1 r_3+\cdots + r_1 r_n) + (r_2r_3 + r_2r_4+\cdots + r_2r_n)+\cdots + r_r_n = \dfrac \\ \quad \vdots \\ r_1 r_2 \dots r_n = (-1)^n \dfrac. \end Vieta's formulas can equivalently be written as : \sum_ \left(\prod_^k r_\right)=(-1)^k\frac for (the indices are sorted in increasing order to ensure each product of roots is used exactly once). The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vector'', commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and relative direction. 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac where , u, is the norm (or length) of u. The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors. Orthogonal coordinates Cartesian coordinates Unit vectors may be us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
