|
Medial 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 w ... [...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] |
Trilinear Coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthologic Triangles
In geometry, two triangles are said to be orthologic if the perpendiculars from the Vertex (geometry), vertices of one of them to the corresponding sides of the other are Concurrent lines, concurrent (i.e., they Line–line intersection, intersect at a single Point (geometry), point). This is a Symmetry (geometry), symmetric property; that is, if the perpendiculars from the vertices of triangle to the sides of triangle are concurrent then the perpendiculars from the vertices of to the sides of are also concurrent. The points of concurrence are known as the orthology centres of the two triangles. Some pairs of orthologic triangles The following are some triangles associated with the reference triangle ABC and orthologic with it. * Medial triangle * Anticomplementary triangle * The triangle whose vertices are the points of contact of the incircle with the sides of ABC * Tangential triangle * The triangle whose vertices are the points of contacts of the excircles with the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inscribed Figure
An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon (or a sphere or ellipsoid inscribed in a convex polyhedron) is tangent to every side or face of the outer figure (but see Inscribed sphere for semantic variants). A polygon inscribed in a circle, ellipse, or polygon (or a polyhedron inscribed in a sphere, ellipsoid, or polyhedron) has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed fig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumconic And Inconic
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Inconic.html Suppose are distinct non-collinear points, and let denote the triangle whose vertices are . Following common practice, denotes not only the vertex but also the angle at vertex , and similarly for and as angles in . Let a= , BC, , b=, CA, , c=, AB, , the sidelengths of . In trilinear coordinates, the general circumconic is the locus of a variable point X = x:y:z satisfying an equation :uyz + vzx + wxy = 0, for some point . The isogonal conjugate of each point on the circumconic, other than , is a point on the line :ux + vy + wz = 0. This line meets the circumcircle of in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle. Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one of the four that does not in general lie on the Euler line. It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers.. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nagel Point
In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurrency of all three of the triangle's splitters. Construction Given a triangle , let be the extouch points in which the -excircle meets line , the -excircle meets line , and the -excircle meets line , respectively. The lines concur in the Nagel point of triangle . Another construction of the point is to start at and trace around triangle half its perimeter, and similarly for and . Because of this construction, the Nagel point is sometimes also called the bisected perimeter point, and the segments are called the triangle's splitters. There exists an easy construction of the Nagel point. Starting from each vertex of a triangle, it suffices to carry twice the length of the opposite edge. We obtain three lines which concur at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nine-point Circle
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of each side of the triangle * The foot of each altitude * The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes). The nine-point circle is also known as Feuerbach's circle (after Karl Wilhelm Feuerbach), Euler's circle (after Leonhard Euler), Terquem's circle (after Olry Terquem), the six-points circle, the twelve-points circle, the -point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the nine-point center of the triangle. Nine Significant Points of Nine Point Circle The diagram above shows the nine significant points of the nine-point circle. Points are the midpoints of the thre ... [...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] |
Euler Line
In geometry, the Euler line, named after Leonhard Euler ( ), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. Triangle centers on the Euler line Individual centers Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. This property is also true for another triangle center, the nine-point center, although it had not been defined in Euler's time. In equilateral triangles, these four points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them. Other notable points that ... [...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] |
