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geometry

 EulerCircle
 find the Euler circle of a given triangle

 Calling Sequence EulerCircle(Elc, T, 'centername'=cn)

Parameters

 T - triangle Elc - the name of the Euler circle 'centername' = cn - (optional) where cn is a name of the center of the Euler's circle.

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{geometry}\right):$
 > $\mathrm{triangle}\left(T,\left[\mathrm{point}\left(A,0,0\right),\mathrm{point}\left(B,2,0\right),\mathrm{point}\left(C,1,3\right)\right]\right):$
 > $\mathrm{EulerCircle}\left(\mathrm{Elc},T,'\mathrm{centername}'=o\right)$
 ${\mathrm{Elc}}$ (1)
 > $\mathrm{detail}\left(\mathrm{Elc}\right)$
 assume that the names of the horizontal and vertical axes are _x and _y, respectively
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{Elc}}\\ {\text{form of the object}}& {\mathrm{circle2d}}\\ {\text{name of the center}}& {o}\\ {\text{coordinates of the center}}& \left[{1}{,}\frac{{5}}{{6}}\right]\\ {\text{radius of the circle}}& \frac{\sqrt{{25}}{}\sqrt{{36}}}{{36}}\\ {\text{equation of the circle}}& {1}{+}{{\mathrm{_x}}}^{{2}}{+}{{\mathrm{_y}}}^{{2}}{-}{2}{}{\mathrm{_x}}{-}\frac{{5}}{{3}}{}{\mathrm{_y}}{=}{0}\end{array}$ (2)
 > $\mathrm{medial}\left(\mathrm{T1},T\right):$
 > $\mathrm{draw}\left(\left\{\mathrm{Elc},T,\mathrm{T1}\right\},\mathrm{printtext}=\mathrm{true}\right)$ 