The Euler circle Elc of triangle T is the circumcircle of the medial triangle of T

Note that it was O. Terquem who named this circle the nine-point circle, and this is the name commonly used in the English-speaking countries. Some French geometers refer to it as Euler's circle, and German geometers usually call it Feuerbach's circle.

If the third optional argument is given and is of the form 'centername' = cn where cn is name, cn will be the name of the center of Elc.

For a detailed description of the Euler circle Elc, use the routine detail (i.e., detail(Elc))

Note that the routine only works if the vertices of triangle T are known.

The command with(geometry,Eulercircle) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{geometry}\right)\:$

$\mathrm{triangle}\left(T\,\left[\mathrm{point}\left(A\,0\,0\right)\,\mathrm{point}\left(B\,2\,0\right)\,\mathrm{point}\left(C\,1\,3\right)\right]\right)\:$

$\mathrm{EulerCircle}\left(\mathrm{Elc}\,T\,'\mathrm{centername}'=o\right)$

$\mathrm{Elc}$

$\mathrm{detail}\left(\mathrm{Elc}\right)$

assume that the names of the horizontal and vertical axes are _x and _y, respectively

$\begin{array}{ll}\text{name of the object}& \mathrm{Elc}\\ \text{form of the object}& \mathrm{circle2d}\\ \text{name of the center}& o\\ \text{coordinates of the center}& \left[1\,\frac{5}{6}\right]\\ \text{radius of the circle}& \frac{\sqrt{25}\sqrt{36}}{36}\\ \text{equation of the circle}& 1+{\mathrm{\_x}}^2+{\mathrm{\_y}}^2-2\mathrm{\_x}-\frac{5}{3}\mathrm{\_y}=0\end{array}$

$\mathrm{medial}\left(\mathrm{T1}\,T\right)\:$

$\mathrm{draw}\left(\left\{\mathrm{Elc}\,T\,\mathrm{T1}\right\}\,\mathrm{printtext}=\mathrm{true}\right)$