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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, isolines of , are circles concentric to circle . Steiner used the power of a point for proofs of several statements on circles, for example: * Determination of a circle, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersecting Chords Theorem
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement 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 and intersecting in a point the following equation holds: , AS, \cdot, SC, =, BS, \cdot, SD, The converse is true as well. That is: If for two line segments and intersecting in the equation above holds true, then their four endpoints lie on a common circle. Or in other words, if the diagonals of a quadrilateral intersect in 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 from the circle's center and is called the ''absolute value of the power of ''; more precisely, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersecting Secants Theorem
In Euclidean geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle. For two lines and that intersect each other at and for which all lie on the same circle, the following equation holds: , PA, \cdot, PD, = , PB, \cdot, PC, The theorem follows directly from the fact that the triangles and are similar. They share and as they are inscribed angles over . 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 In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengt ... and the tangent-secant theorem, the intersec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cosines
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see Fig. 1), the law of cosines states: \begin c^2 &= a^2 + b^2 - 2ab\cos\gamma, \\[3mu] a^2 &= b^2+c^2-2bc\cos\alpha, \\[3mu] b^2 &= a^2+c^2-2ac\cos\beta. \end The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if is a right angle then , and the law of cosines special case, reduces to . The law of cosines is useful for solution of triangles, solving a triangle when all three sides or two sides and their included angle are given. Use in solving triangles The theorem is used in solution of triangles, i.e., to find (see Figure 3): *the third side of a triangle if two sides and the angle between them is known: c = \sqrt\,; *the angles of a triangle if the three sides are known: \gamma = \arccos\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kreis is the German word for circle. Kreis may also refer to: Places * , or Circle (administrative division), 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 ** ''Kreis'' (Habsburg monarchy), a former type of subdivision of the Habsburg monarchy and Austrian Empire *, 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 divisi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similarity (geometry)
In Euclidean geometry, two objects are similar if they have the same shape, or if 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. This is because two ellipse ... [...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 called its ''center'' and a nonzero number called its ''ratio'', which sends point to a point 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 translations, all homotheties of an affine (or Euclid ... [...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 (1540-1603), more commonly referred to by the Latinised form of his name, "Franciscus Vieta." Basic formulas Any general polynomial of degree ''n'' P(x) = a_n x^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 coefficients to signed sums of products of the roots as follows: 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 left-hand sides of Vieta's formulas are the elementary symmetric polynomials of the roots. Vieta's system can be solved by Newton's method through an explicit simple iterative formula, the Durand- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), 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 ''normalized vector'' is sometimes used as a synonym for ''unit vector''. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac=(\frac, \frac, ... , \frac) where ‖u‖ is the Norm (mathematics), norm (or length) of u and \, \mathbf\, = (u_1, u_2, ..., u_n). The proof is the following: \, \mathbf\, =\sqrt=\sqrt=\sqrt=1 A unit vector is often used to represent direction (geometry), directions, such as normal directions. Unit vectors are often chosen to form the basis (linear algebra), basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors. Orthogonal coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
