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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's position depends on the type of triangle: For an acute triangle (all angles smaller than a right angle), the circumcenter always lies inside the triangle. For a right triangle, the circumcenter always lies at the midpoint of the hypotenuse. This is one form of Thales' theorem.
X(2) Centroid: intersection of the three medians: X(3) Circumcenter: center of the circumscribed circle: X(4) orthocenter: intersection of the three altitudes: X(5) nine-point center: center of the nine-point circle: X(6) symmedian point: intersection of the three symmedians: X(7) Gergonne point: symmedian point of contact triangle X(8) Nagel point
The medial triangle of the intouch triangle is inverted into triangle ABC, meaning the circumcenter of the medial triangle, that is, the nine-point center of the intouch triangle, the incenter and circumcenter of triangle ABC are collinear. Any two non-intersecting circles may be inverted into concentric circles.
The three perpendicular bisectors meet in a single point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. [20] Thales' theorem implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle. [21]
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.
If the sidelengths of triangle ABC are a, b, c the baricentric coordinates of the Lemoine point are a 2 : b 2 : c 2. It has been described as "one of the crown jewels of modern geometry". [9] There are several earlier references to this point in the mathematical literature details of which are available in John Mackay' history of the symmedian ...
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]