Let circle $(F)$ and parabola $\mathcal{F}$ with focus $F$ meet at $X,Y.$ Let $A,B,C,D$ be four points on $(F)$ (not lying on its arc XY cut by the angle XFY) such that $AB,BC,CD$ are tangent to $\mathcal{F}.$ Denote $P \equiv AB \cap CD.$ Reflections $F_1,F_2,F_3$ of focus $F$ on tangents $PB,$ $PC,$ $BC$ lie on the directrix $\tau$ of $\mathcal{F}$ $\Longrightarrow$ $F \in \odot(PBC),$ this is $\tau \equiv F_1F_2F_3$ is the Steiner line of $F$ with respect to $\triangle PBC.$ Consequently, $\angle BPC=\angle BFC=2\angle BAC$ $\Longrightarrow$ $\triangle APC$ is isosceles with apex $P.$ Thus, $ADBC$ is an isosceles trapezoid with $AC \parallel BD.$ By axial symmetry, $F \in \odot(PAD)$ $\Longrightarrow$ $\tau$ is also the Steiner line of $F$ WRT $\triangle PAD,$ i.e. Reflection $F_4$ of $F$ across $DA$ lies on $\tau$ $\Longrightarrow$ $DA$ is tangent to $\mathcal{F}.$