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Brocard Points
In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle with sides , where the vertices are labeled in counterclockwise order, there is exactly one point such that the line segments form the same angle, , with the respective sides , namely that \angle PAB = \angle PBC = \angle PCA =\omega.\, Point is called the first Brocard point of the triangle , and the angle is called the Brocard angle of the triangle. This angle has the property that \cot\omega = \cot\!\bigl(\angle CAB\bigr) + \cot\!\bigl(\angle ABC\bigr) + \cot\!\bigl(\angle BCA\bigr). There is also a second Brocard point, , in triangle such that line segments form equal angles with sides respectively. In other words, the equations \angle QCB = \angle QBA = \angle QAC apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words, angle \angle PBC = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anticomplementary Triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is not the same thing as the median triangle, which is the triangle whose sides have the same lengths as the medians of . Each side of the medial triangle is called a ''midsegment'' (or ''midline''). In general, a midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to half the length of the third side. Properties The medial triangle can also be viewed as the image of triangle transformed by a homothety centered at the centroid with ratio -1/2. Thus, the sides of the medial triangle are half and parallel to the corresponding sides of triangle ABC. Hence, the medial triangle is inversely similar and shares the same centroid and medians wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore, two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pedal Triangle
In plane geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle , and a point that is not one of the vertices . Drop perpendiculars from to the three sides of the triangle (these may need to be produced, i.e., extended). Label the intersections of the lines from with the sides . The pedal triangle is then . If is not an obtuse triangle and is the orthocenter, then the angles of are , and . The quadrilaterals are cyclic quadrilaterals. The location of the chosen point relative to the chosen triangle gives rise to some special cases: * If is the orthocenter, then is the orthic triangle. * If is the incenter, then is the intouch triangle. * If is the circumcenter, then is the medial triangle. *If is on the circumcircle of the triangle, collapses to a line (the ''pedal line'' or ''Simson line''). The vertices of the pedal triangle of an interior point , as shown in the top diag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length d of a diameter is also called the diameter. In this sense one speaks of diameter rather than diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r. :d = 2r \qquad\text\qquad r = \frac. The word "diameter" is derived from (), "diameter of a circle", from (), "across, through" and (), "measure". It is often abbreviated \text, \text, d, or \varnothing. Constructions With straightedge and compass, a diameter of a given circle can be constructed as the perpendicular bisector of an arbitrary chord. Drawing two diameters in this way can be used to locate the center of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concyclic
In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertices are concyclic is called a cyclic polygon, and the circle is called its ''circumscribing circle'' or ''circumcircle''. All concyclic points are equidistant from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, so every triangle is a cyclic polygon, with a well-defined circumcircle. However, four or more points in the plane are not necessarily concyclic. After triangles, the special case of cyclic quadrilaterals has been most extensively studied. Perpendicular bisectors In general the centre ''O'' of a circle on which points ''P'' and ''Q'' lie must be such that ''OP'' and ''OQ'' are equal distances. Therefore ''O'' must lie on the perpendicular bisector of the line segment ''PQ''. For ''n'' distinct points there are ''n''(''n'' − 1)/2 bisectors, and the concyclic condition is that t ... [...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. In other words, it is the isogonal conjugate of the centroid. 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 punctured at its own center, and could be any point therein. The symmedian point of a triangle with side lengths , and has homogene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcenter
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case , a cyclic quadrilateral. All rectangles, isosceles trapezoids, right kites, and regular polygons are cyclic, but not every polygon is. Straightedge and compass construction The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. For three non-collinear points, these two lines cannot be parallel, and the circumcenter is the point where they cross. Any point on the bisector is equidistant from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular Bisector
In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'', a line that passes through the midpoint of a given segment, and the ''angle bisector'', a line that passes through the apex of an angle (that divides it into two equal angles). In three-dimensional space, bisection is usually done by a bisecting plane, also called the ''bisector''. Perpendicular line segment bisector Definition *The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly. *The perpendicular bisector of a line segment AB also has the property that each of its points X is equidistant from segment AB's endpoints: (D)\quad , XA, = , XB, . The proof follows from , MA, =, MB, and Pythagoras' theorem: :, XA, ^2=, XM, ^2+, MA, ^2=, XM, ^2+, MB, ^2=, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcircle
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection (geometry), intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case , a cyclic quadrilateral. All rectangles, isosceles trapezoids, right kites, and regular polygons are cyclic, but not every polygon is. Straightedge and compass construction The circumcenter of a triangle can be Compass-and-straightedge construction, constructed by drawing any two of the three Bisection#Perpendicular bisectors, perpendicular bisectors. For three non-collinear points, these two lines cannot be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmedian Point
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector. The three symmedians meet at a triangle center called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry".. Isogonality Many times in geometry, if we take three special lines through the vertices of a triangle, or '' cevians'', then their reflections about the corresponding angle bisectors, called ''isogonal lines'', will also have interesting properties. For instance, if three cevians of a triangle inters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotomic Conjugate
In geometry, the isotomic conjugate of a point with respect to a triangle is another point, defined in a specific way from and : If the base points of the lines on the sides opposite are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of . Construction We assume that is not collinear with any two vertices of . Let be the points in which the lines meet sidelines ( extended if necessary). Reflecting in the midpoints of sides will give points respectively. The isotomic lines joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the ''isotomic conjugate'' of . Coordinates If the trilinears for are , then the trilinears for the isotomic conjugate of are :a^p^ : b^q^ : c^r^, where are the side lengths opposite vertices respectively. Properties The isotomic conjugate of the centroid of triangle is the centroid itself. The isotomic conjugate of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
