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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).
By Euler's theorem in geometry, the distance between the circumcenter O and the incenter I is O I ¯ = R ( R − 2 r ) , {\displaystyle {\overline {OI}}={\sqrt {R(R-2r)}},} where r is the incircle radius and R is the circumcircle radius; hence the circumradius is at least twice the inradius ( Euler's triangle inequality ), with equality only in ...
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 ]
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 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 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.
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.
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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