Fuhrmann triangle

The Fuhrmann triangle, named after Wilhelm Fuhrmann (1833–1904), is special triangle based on a given arbitrary triangle.

Fuhrmann triangle (red): ${\displaystyle \triangle M_{c}^{\prime }M_{b}^{\prime }M_{a}^{\prime }}$
mid arc points: ${\displaystyle M_{a},M_{b},M_{c}}$

Fuhrmann triangle (red): ${\displaystyle \triangle M_{c}^{\prime }M_{b}^{\prime }M_{a}^{\prime }}$
${\displaystyle \triangle M_{c}^{\prime }M_{b}^{\prime }M_{a}^{\prime }\sim \triangle M_{a}M_{b}M_{c}}$

For a given triangle ${\displaystyle \triangle ABC}$ and its circumcircle the midpoints of the arcs over triangle sides are denoted by ${\displaystyle M_{a},M_{b},M_{c}}$. Those midpoints get reflected at the associated triangle sides yielding the points ${\displaystyle M_{a}^{\prime },M_{b}^{\prime },M_{c}^{\prime }}$, which forms the Fuhrmann triangle.^[1][2]

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 ${\displaystyle \triangle M_{c}^{\prime }M_{b}^{\prime }M_{a}^{\prime }\sim \triangle M_{a}M_{b}M_{c}}$.^[1] For the area of the Fuhrmann triangle the following formula holds:^[3]

${\displaystyle |\triangle M_{c}^{\prime }M_{b}^{\prime }M_{a}^{\prime }|={\frac {(a+b+c)|OI|^{2}}{4R}}={\frac {(a+b+c)(R-2r)}{4}}}$

Where ${\displaystyle O}$ denotes the circumcenter of the given triangle ${\displaystyle \triangle ABC}$ and ${\displaystyle R}$ its radius as well as ${\displaystyle I}$ denoting the incenter and ${\displaystyle r}$ its radius. Due to Euler's theorem one also has ${\displaystyle |OI|^{2}=R(R-2r)}$. The following equations hold for the sides of the Fuhrmann triangle:^[3]

${\displaystyle a^{\prime }={\sqrt {\frac {(-a+b+c)(a+b+c)}{bc}}}|OI|}$

${\displaystyle b^{\prime }={\sqrt {\frac {(a-b+c)(a+b+c)}{ac}}}|OI|}$

${\displaystyle c^{\prime }={\sqrt {\frac {(a+b-c)(a+b+c)}{ab}}}|OI|}$

Where ${\displaystyle a,b,c}$ denote the sides of the given triangle ${\displaystyle \triangle ABC}$ and ${\displaystyle a^{\prime },b^{\prime },c^{\prime }}$ the sides of the Fuhrmann triangle (see drawing).