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Examples of cyclic quadrilaterals. In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle.This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
The circumcircle of three collinear points is the line on which the three points lie, often referred to as a circle of infinite radius. Nearly collinear points often lead to numerical instability in computation of the circumcircle. Circumcircles of triangles have an intimate relationship with the Delaunay triangulation of a set of points.
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.
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.
The orthocenter (blue point), the center of the nine-point circle (red), the centroid (orange), and the circumcenter (green) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter ...
It has also rarely been called a double circle quadrilateral [2] and double scribed quadrilateral. [3] If two circles, one within the other, are the incircle and the circumcircle of a bicentric quadrilateral, then every point on the circumcircle is the vertex of a bicentric quadrilateral having the same incircle and circumcircle. [4]
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.
Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on ...