Two circles c1 and c2 are said to be orthogonal if the angles of intersection of the two circles are right angles. By the angles of intersection of two circle (or two coplanar curves in general) at a point which they have in common is meant that the angles between the tangents to the curves at the common point.

The routine returns true if c1 and c2 are orthogonal; false if they are not; and FAIL if it is unable to reach a conclusion.

In case of FAIL, if the third optional argument is given, the condition that makes c1 and c2 orthogonal to each other is assigned to this argument.

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

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

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

$\mathrm{circle}\left(\mathrm{c1}\,x^2+y^2=1\right),\mathrm{circle}\left(\mathrm{c2}\,{\left(x-2\right)}^2+y^2=2\right)\:$

$\mathrm{circle}\left(\mathrm{c3}\,x^2+y^2=2\right)\:$

$\mathrm{AreOrthogonal}\left(\mathrm{c1}\,\mathrm{c2}\right)$

$\mathrm{false}$

$\mathrm{AreOrthogonal}\left(\mathrm{c2}\,\mathrm{c3}\right)$

$\mathrm{true}$