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The center of the incircle is a triangle center called the triangle's incenter. [1] An excircle or escribed circle [2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. [3]
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 .
It is the first listed center, X(1), in Clark Kimberling's Encyclopedia of Triangle Centers, and the identity element of the multiplicative group of triangle centers. [ 1 ] [ 2 ] For polygons with more than three sides, the incenter only exists for tangential polygons : those that have an incircle that is tangent to each side of the polygon.
That is, the Spieker center of ABC is the center of the circle inscribed in the medial triangle of ABC. This circle is known as the Spieker circle. The Spieker center is also located at the intersection of the three cleavers of triangle ABC. A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint
The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center.
According to Lester's theorem, the nine-point center lies on a common circle with three other points: the two Fermat points and the circumcenter. [9] The Kosnita point of a triangle, a triangle center associated with Kosnita's theorem, is the isogonal conjugate of the nine-point center. [10]
In geometry, the Bevan point, named after Benjamin Bevan, is a triangle center. It is defined as center of the Bevan circle, that is the circle through the centers of the three excircles of a triangle. [1] The Bevan point of a triangle is the reflection of the incenter across the circumcenter of the triangle. [1]
where A, B, C denote both the triangle's vertices and the angle measures at those vertices; H is the orthocenter (the intersection of the triangle's altitudes); D, E, F are the feet of the altitudes from vertices A, B, C respectively; R is the triangle's circumradius (the radius of its circumscribed circle); and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C ...