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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 synthetic affine definition of the midpoint M of a segment AB is the projective harmonic conjugate of the point at infinity, P, of the line AB. That is, the point M such that H[A,B; P,M]. [6] When coordinates can be introduced in an affine geometry, the two definitions of midpoint will coincide. [7]
The Simson line LN (red) of the triangle ABC with respect to point P on the circumcircle. In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. [1] The line through these points is the Simson line of P, named for Robert Simson. [2]
Let be the intersection of line with and be the intersection of line with . Homothety with center on T A {\displaystyle T_{A}} between Ω A {\displaystyle \Omega _{A}} and Γ {\displaystyle \Gamma } implies that X , Y {\displaystyle X,Y} are the midpoints of Γ {\displaystyle \Gamma } arcs A B {\displaystyle AB} and A C {\displaystyle AC ...
In geometry, the Euler line, named after Leonhard Euler (/ ˈ ɔɪ l ər / OY-lər), 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.
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction: Given three collinear points A, B, C, let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively.
The common line or line segment for the midpoints is called the diameter. For a circle , ellipse or hyperbola the diameter goes through its center . For a parabola the diameter is always perpendicular to its directrix and for a pair of intersecting lines (from a degenerate conic ) the diameter goes through the point of intersection.
In algebraic geometry, a line complex is a 3-fold given by the intersection of the Grassmannian G(2, 4) (embedded in projective space P 5 by Plücker coordinates) with a hypersurface. It is called a line complex because points of G (2, 4) correspond to lines in P 3 , so a line complex can be thought of as a 3-dimensional family of lines in P 3 .