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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
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 .
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:
The distance from the incenter to the centroid is less than one third the length of the longest median of the triangle. [9] By Euler's theorem in geometry, the squared distance from the incenter I to the circumcenter O is given by [10] [11] = (),
The nine-point center is the circumcenter of the medial triangle of the given triangle, the circumcenter of the orthic triangle of the given triangle, and the circumcenter of the Euler triangle. [3] More generally it is the circumcenter of any triangle defined from three of the nine points defining the nine-point circle.
If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side. [23] In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect. [23]
and the distance from the incenter to the center of the nine point circle is [17]: 232 ¯ = <. The incenter lies in the medial triangle (whose vertices are the midpoints of the sides). [ 17 ] : 233, Lemma 1
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