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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).
Fuss' theorem gives a relation between the inradius r, the circumradius R and the distance x between the incenter I and the circumcenter O, for any bicentric quadrilateral. The relation is [1] [11] [22] + (+) =, or equivalently
By Euler's theorem in geometry, the squared distance from the incenter I to the circumcenter O is given by [10] [11] O I 2 = R ( R − 2 r ) , {\displaystyle OI^{2}=R(R-2r),} where R and r are the circumradius and the inradius respectively; thus the circumradius is at least twice the inradius, with equality only in the equilateral case.
the circumcentre, which is the centre of the circle that passes through all three vertices; the centroid or centre of mass, the point on which the triangle would balance if it had uniform density; the incentre, the centre of the circle that is internally tangent to all three sides of the triangle;
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.
which is also the distance between the circumcenter and incenter. [2] Aside from the orthocenter the Fuhrmann circle intersects each altitude of the triangle in one additional point. Those points all have the distance from their associated vertices of the triangle.
The radius of the inscribed circle is the apothem (the shortest distance from the center to the boundary of the regular polygon). For any regular polygon, the relations between the common edge length a, the radius r of the incircle, and the radius R of the circumcircle are:
Power of a point – Relative distance of a point from a circle; Steiner inellipse – Unique ellipse tangent to all 3 midpoints of a given triangle's sides; Tangential quadrilateral – Polygon whose four sides all touch a circle; Triangle conic; Incenter–excenter lemma – A statement about properties of inscribed and circumscribed circles