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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 .
Such a circle is said to circumscribe the points or a polygon formed from them; such a polygon is said to be inscribed in the circle. Circumcircle, the circumscribed circle of a triangle, which always exists for a given triangle. Cyclic polygon, a general polygon that can be circumscribed by a circle. The vertices of this polygon are concyclic ...
A sequence of circumscribed polygons and circles. Any regular polygon is cyclic. Consider a unit circle, then circumscribe a regular triangle such that each side touches the circle. Circumscribe a circle, then circumscribe a square. Again circumscribe a circle, then circumscribe a regular pentagon, and so on.
Center of the triangle's inscribed circle. X 2: Centroid: G:: Intersection of the medians. Center of mass of a uniform triangular lamina. X 3: Circumcenter: O : : Intersection of the perpendicular bisectors of the sides. Center of the triangle's circumscribed circle. X 4: Orthocenter: H
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).
In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle. [19] About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices. [20]
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 center of the circle and its radius are called the circumcenter and the circumradius respectively.
Each hyperbolic triangle has an inscribed circle but not every hyperbolic triangle has a circumscribed circle (see below). Its vertices can lie on a horocycle or hypercycle. Hyperbolic triangles have some properties that are analogous to those of triangles in spherical or elliptic geometry: Two triangles with the same angle sum are equal in area.