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Feuerbach Hyperbola
In geometry, the Feuerbach hyperbola is a rectangular hyperbola passing through important triangle centers such as the Orthocenter, Gergonne point, Nagel point and Shiffler point. The center of the hyperbola is the Feuerbach point, the point of tangency of the incircle and the nine-point circle. Equation It has the trilinear equation \frac+ \frac+ \frac(here A,B,C are the angles at the respective vertices and (\alpha,\beta,\gamma) is the barycentric coordinate). Properties Like other rectangular hyperbolas, the orthocenter of any three points on the curve lies on the hyperbola. So, the orthocenter of the triangle ABC lies on the curve. The line OI is tangent to this hyperbola at I. Isogonal conjugate of OI The hyperbola is the isogonal conjugate of OI, the line joining the circumcenter and the incenter. This fact leads to a few interesting properties. Specifically all the points lying on the line OI have their isogonal conjugates lying on the hyperbola. The Nagel po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixtilinear Incircles Of A Triangle
In geometry, a mixtilinear incircle of a triangle is a circle tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex A is called the ''A-mixtilinear incircle.'' Every triangle has three unique mixtilinear incircles, one corresponding to each vertex. Proof of existence and uniqueness The A-excircle of triangle ABC is unique. Let \Phi be a transformation defined by the composition of an inversion centered at A with radius \sqrt and a reflection with respect to the angle bisector on A. Since inversion and reflection are bijective and preserve touching points, then \Phi does as well. Then, the image of the A-excircle under \Phi is a circle internally tangent to sides AB, AC and the circumcircle of ABC, that is, the A-mixtilinear incircle. Therefore, the A-mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcenter
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are concyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it, if the circle's center is within the polygon. Every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm. Even if a polygon has a circumscribed circle, it may be different from its minimum bounding circle. For example, for an obtuse triangle, the minimum bounding circle has the longest s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Václav Jeřábek
Václav Jeřábek (1845–1931) was a Czech mathematician, specialized in constructive geometry. Life and work Jeřábek studied at the lower school of Pardubice and at the higher school of Písek, then he was to Vienna and studied at Imperial and Royal Polytechnic Institute where he graduated. Although he participated in several leading intellectual circles of Vienna, he remained a Czech with a clear view of patriotism. He began his teaching at the ''Realschule'' of Litomyšl (1870), being transferred two years after to the ''Realschule'' of Telč. In 1881, he was appointed professor of the ''Czech Realschule'' in Brno, and became its director in 1901. He retired in 1907, and suffering of a cataract, he died almost completely blind, MacTutor History of Mathematics. in 1931. Jeřábek was one of the men who kept the Czech geometry at the scientific level. He published scientific articles in Czech, German and French, and longer lectures. He is well remembered by the Jerabek hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ''extended base'' of the altitude. The intersection of the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kiepert Conics
In triangle geometry, the Kiepert conics are two special conics associated with the reference triangle. One of them is a hyperbola, called the Kiepert hyperbola and the other is a parabola, called the Kiepert parabola. The Kiepert conics are defined as follows: :If the three triangles A^\prime BC, AB^\prime C and ABC^\prime, constructed on the sides of a triangle ABC as bases, are similar, isosceles and similarly situated, then the triangles ABC and A^\prime B^\prime C^\prime are in perspective. As the base angle of the isosceles triangles varies between -\pi/2 and \pi/2, the locus of the center of perspectivity of the triangles ABC and A^\prime B^\prime C^\prime is a hyperbola called the Kiepert hyperbola and the envelope of their axis of perspectivity is a parabola called the Kiepert parabola. It has been proved that the Kiepert hyperbola is the hyperbola passing through the vertices, the centroid and the orthocenter of the reference triangle and the Kiepert parabola is the pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi's Theorem (geometry)
In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle ''ABC'' and a triple of angles ''α'', ''β'', and ''γ''. This information is sufficient to determine three points ''X'', ''Y'', and ''Z'' such that ∠''ZAB'' = ∠''YAC'' = ''α'', ∠''XBC'' = ∠''ZBA'' = ''β'', and ∠''YCA'' = ∠''XCB'' = ''γ''. Then, by a theorem of , the lines ''AX, BY,'' and ''CZ'' are concurrent,Glenn T. Vickers, "Reciprocal Jacobi Triangles and the McCay Cubic", ''Forum Geometricorum 15'', 2015, 179–183. http://forumgeom.fau.edu/FG2015volume15/FG201518.pdfGlenn T. Vickers, "The 19 Congruent Jacobi Triangles", ''Forum Geometricorum'' 16, 2016, 339–344. http://forumgeom.fau.edu/FG2016volume16/FG201642.pdf at a point ''N'' called the Jacobi point. The Jacobi point is a generalization of the Fermat point, which is obtained by letting ''α'' = ''β'' = ''γ'' = 60° and triangle ''ABC'' having no angle being greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemoine Point
In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians ( medians reflected at the associated angle bisectors) of a triangle. Ross Honsberger called its existence "one of the crown jewels of modern geometry". In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X(6).Encyclopedia of Triangle Centers accessed 2014-11-06. For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein. The symmedian point of a triangle with side lengths , and has homogeneous |
Pedal Triangle
In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the three sides of the triangle (these may need to be produced, i.e., extended). Label ''L'', ''M'', ''N'' the intersections of the lines from ''P'' with the sides ''BC'', ''AC'', ''AB''. The pedal triangle is then ''LMN''. If ABC is not an obtuse triangle, P is the orthocenter then the angles of LMN are 180°−2A, 180°−2B and 180°−2C. The location of the chosen point ''P'' relative to the chosen triangle ''ABC'' gives rise to some special cases: * If ''P = '' orthocenter, then ''LMN = '' orthic triangle. * If ''P = ''incenter, then ''LMN = ''intouch triangle. * If ''P = '' circumcenter, then ''LMN = '' medial triangle. If ''P'' is on the circumcircle of the triangle, ''LMN'' collapses to a line. This is then called the pedal line, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homothetic Center
In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another. If the center is ''external'', the two figures are directly similar to one another; their angles have the same rotational sense. If the center is ''internal'', the two figures are scaled mirror images of one another; their angles have the opposite sense. General polygons If two geometric figures possess a homothetic center, they are similar to one another; in other words, they must have the same angles at corresponding points and differ only in their relative scaling. The homothetic center and the two figures need not lie in the same plane; they can be related by a projection from the homothetic center. Homothetic centers may be external or internal. If the center is internal, the two geometric figures are scaled mirror images of one another; in technica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycentric Coordinate System
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane (mathematics), plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or ''barycenter'') of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex. Every point has barycentric coordinates, and their sum is not zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity. Bar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
