Euler's theorem in geometry

In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by^{[1] [2]} $${\displaystyle d^{2}=R(R-2r)}$${\displaystyle d^{2}=R(R-2r)} or equivalently $${\displaystyle {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},}$${\displaystyle {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},} where ${\displaystyle R}${\displaystyle R} and ${\displaystyle r}${\displaystyle r} denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively). The theorem is named for Leonhard Euler, who published it in 1765.^[3] However, the same result was published earlier by William Chapple in 1746.^[4]

From the theorem follows the Euler inequality:^[5] $${\displaystyle R\geq 2r,}$${\displaystyle R\geq 2r,} which holds with equality only in the equilateral case.^[6]

A stronger version^[6] is $${\displaystyle {\frac {R}{r}}\geq {\frac {abc+a^{3}+b^{3}+c^{3}}{2abc}}\geq {\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}-1\geq {\frac {2}{3}}\left({\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}\right)\geq 2,}$${\displaystyle {\frac {R}{r}}\geq {\frac {abc+a^{3}+b^{3}+c^{3}}{2abc}}\geq {\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}-1\geq {\frac {2}{3}}\left({\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}\right)\geq 2,} where ${\displaystyle a}${\displaystyle a}, ${\displaystyle b}${\displaystyle b}, and ${\displaystyle c}${\displaystyle c} are the side lengths of the triangle.

If ${\displaystyle r_{a}}${\displaystyle r_{a}} and ${\displaystyle d_{a}}${\displaystyle d_{a}} denote respectively the radius of the escribed circle opposite to the vertex ${\displaystyle A}${\displaystyle A} and the distance between its center and the center of the circumscribed circle, then ${\displaystyle d_{a}^{2}=R(R+2r_{a})}${\displaystyle d_{a}^{2}=R(R+2r_{a})}.

Euler's inequality, in the form stating that, for all triangles inscribed in a given circle, the maximum of the radius of the inscribed circle is reached for the equilateral triangle and only for it, is valid in absolute geometry.^[7]

• Fuss' theorem for the relation among the same three variables in bicentric quadrilaterals
• Poncelet's closure theorem, showing that there is an infinity of triangles with the same two circles (and therefore the same R, r, and d)
• Egan conjecture, generalization to higher dimensions
• List of triangle inequalities

• Weisstein, Eric W., "Euler Triangle Formula", MathWorld

• Triangle inequalities
• Theorems about triangles and circles

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