The routine tests if the four given points P1, P2, P3, and P4 are concyclic, i.e., if they lie on the same circle. It returns true if they are; false if they are not; or FAIL if it is able to determine if they are concyclic.

If FAIL is returned, and the optional argument cond is given, the condition that makes the points concyclic is assigned to this argument.

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

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

$\mathrm{point}\left(\mathrm{P1}\,0\,0\right),\mathrm{point}\left(\mathrm{P2}\,2\,0\right),\mathrm{point}\left(\mathrm{P3}\,2\,2\right)\:$

$\mathrm{point}\left(\mathrm{P4}\,0\,2\right),\mathrm{point}\left(\mathrm{P5}\,1\,7\right)\:$

$\mathrm{AreConcyclic}\left(\mathrm{P1}\,\mathrm{P2}\,\mathrm{P3}\,\mathrm{P4}\right)$

$\mathrm{true}$

$\mathrm{AreConcyclic}\left(\mathrm{P1}\,\mathrm{P2}\,\mathrm{P3}\,\mathrm{P5}\right)$

$\mathrm{false}$

$\mathrm{point}\left(\mathrm{P5}\,a\,b\right)\:$

$\mathrm{AreConcyclic}\left(\mathrm{P1}\,\mathrm{P2}\,\mathrm{P3}\,\mathrm{P5}\,'\mathrm{cond}'\right)$

AreConcyclic:   "unable to determine if 32/45*(a^2+b^2-2*a-2*b)/(a^2+b^2+1) is zero"

$\mathrm{FAIL}$

$\mathrm{cond}$

$\frac{32\left(a^2+b^2-2a-2b\right)}{45\left(a^2+b^2+1\right)}=0$

$\mathrm{assume}\left(\mathrm{cond}\right)$

$\mathrm{AreConcyclic}\left(\mathrm{P1}\,\mathrm{P2}\,\mathrm{P3}\,\mathrm{P5}\right)$

$\mathrm{true}$