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 formula_1 or equivalently formula_2 where formula_3 and formula_4 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. However, the same result was published earlier by William Chapple in 1746. From the theorem follows the Euler inequality: formula_5 which holds with equality only in the equilateral case. Stronger version of the inequality. A stronger version is  formula_6 where formula_7, formula_8, and formula_9 are the side lengths of the triangle. Euler's theorem for the escribed circle. If formula_10 and formula_11 denote respectively the radius of the escribed circle opposite to the vertex formula_12 and the distance between its center and the center of the circumscribed circle, then formula_13. Euler's inequality in absolute geometry. 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. Euler's