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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 circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an n -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 n = 4 , a cyclic quadrilateral .
Common nine-point circle, where N, O 4, A 4 are the nine-point center, circumcenter, and orthocenter respectively of the triangle formed from the other three orthocentric points A 1, A 2, A 3. The center of this common nine-point circle lies at the centroid of the four orthocentric points. The radius of the common nine-point circle is the ...
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.
A triangle showing its circumcircle and circumcenter (black), altitudes and orthocenter (red), and nine-point circle and nine-point center (blue) In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle.
As the reflection of the orthocenter around the circumcenter, the de Longchamps point belongs to the line through both of these points, which is the Euler line of the given triangle. Thus, it is collinear with all the other triangle centers on the Euler line, which along with the orthocenter and circumcenter include the centroid and the center ...
Likewise, a triangle's circumcenter—the intersection of the three sides' perpendicular bisectors, which is the center of the circle that passes through all three vertices—falls inside an acute triangle but outside an obtuse triangle. The right triangle is the in-between case: both its circumcenter and its orthocenter lie on its boundary.
Each point in the list is identified by an index number of the form X(n) —for example, X(1) is the incenter. [6] The information recorded about each point includes its trilinear and barycentric coordinates and its relation to lines joining other identified points. Links to The Geometer's Sketchpad diagrams are provided for key points. The ...
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