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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.
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 orthopole of lines passing through the circumcenter lie on the nine-point circle. A triangle's circumcircle, its nine-point circle, its polar circle, and the circumcircle of its tangential triangle [9] are coaxal. [10] Trilinear coordinates for the center of the Kiepert hyperbola are
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.
The vertices of every triangle fall on a circle called the circumcircle. (Because of this, some authors define "concyclic" only in the context of four or more points on a circle.) [2] Several other sets of points defined from a triangle are also concyclic, with different circles; see Nine-point circle [3] and Lester's theorem.
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.
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: intersection of lines from each vertex to the corresponding semiperimeter point X(9) Mittenpunkt