As per the definition given on this site,
Given an origin O and a point P on the curve, let B be the point where the extension of the line OP intersects the line $x=2a$ and C be the intersection of the circle of radius $a$ and center $(a,0)$ with the extension of OP. Then the Cissoid of Diocles is the curve which satisfies OP=CB.
But is this condition applicable to the points on the curve that lie outside the circle? Clearly, if point 'P' was outside the circle, then the extension of OP wouldn't have intersected the circle.
What is it that I'm missing here?
$OP$ is considered part of its own extension, so for $P$ outside the circle, $C$ is between $O$ and $P$ (and the required equality may be restated as $OC=PB$). There is thus no problem in the definition.