Fuhrmann triangle
The Fuhrmann triangle, named after Wilhelm Fuhrmann (1833–1904), is special triangle based on a given arbitrary triangle. For a given triangle formula_1 and its circumcircle the midpoints of the arcs over triangle sides are denoted by formula_2. Those midpoints get reflected at the associated triangle sides yielding the points formula_3, which forms the "Fuhrmann triangle". The circumcircle of Fuhrmann triangle is the Fuhrmann circle. Furthermore the Furhmann triangle is similar to the triangle formed by the mid arc points, that is formula_4. For the area of the Fuhrmann triangle the following formula holds: Where formula_6 denotes the circumcenter  of the given triangle formula_1 and formula_8 its radius as well as formula_9 denoting the incenter and formula_10 its radius. Due to Euler's theorem one also has formula_11. The following equations hold for the sides of the Fuhrmann triangle: Where formula_15 denote the sides of the given triangle formula_1 and formula_17 the sides of the Fuhrmann triangle (see drawing).