The pole of a line p with respect to a circle c, center O is the inverse of foot of perpendicular from O to p.  Alternatively, the point of intersection of the polars of any two points on p.

For a detailed description of the pole of P, use the routine detail.

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

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

$\mathrm{\_EnvHorizontalName}≔'x'\:$$\mathrm{\_EnvVerticalName}≔'y'\:$

$\mathrm{circle}\left(c\,x^2+y^2=1\right),\mathrm{ellipse}\left(e\,\frac{x^2}{4}+y^2=1\right)$

$c,e$

$\mathrm{line}\left(\mathrm{l1}\,3x-1=0\right),\mathrm{line}\left(\mathrm{l2}\,\frac{3}{4}x-1=0\right)$

$\mathrm{l1},\mathrm{l2}$

$\mathrm{Pole}\left(\mathrm{P1}\,\mathrm{l1}\,c\right)$

$\mathrm{P1}$

$\mathrm{coordinates}\left(\mathrm{P1}\right)$

$\left[3\,0\right]$

$\mathrm{Pole}\left(\mathrm{P2}\,\mathrm{l2}\,e\right)$

$\mathrm{P2}$

$\mathrm{coordinates}\left(\mathrm{P2}\right)$

$\left[3\,0\right]$