A plane curve, $α(θ)$, has the following property: if $c(θ)$ is the center of curvature of $α$ in $θ$, $Q(θ)$ is the projection of $α(θ)$ on the x axis and $T(θ)$ is the intersection point of the tangent line to $α$ in $θ$ with the x axis, then the area of the triangle $cQT$ is constant. Give the parametrization for the curve $α(θ)$, where the parameter $θ$ is the angle between the tangent to $α(θ)$ and the x axis.
I know $c(θ) = α(θ) + \frac{n(θ)}{k(θ)}$, and $Q(θ) = (x(θ), 0)$ (given that $α(θ) = (x(θ),y(θ))$, but I haven't been able to put all the other information together. I know I should find an expression for $T(θ)$ as well, but I'm having trouble with that too. I've been stuck on this for a while, so I would appreciate being pointed in the right direction.
EDIT: After over a year with no progress to this question, I'm giving it a bounty. It's eating away at me that I haven't been able to solve such a basic problem.