I am given: Five points on the ellipse, The sharing center of the ellipse and the circle, And the circle.
One of those five points is one of the intersections of a circle and an ellipse.
Can we construct the three intersections of an ellipse and a circle that is not given?
Blue ones are given, and greens are the points I want to find.
Let $ABCDE$ be the five given points on the ellipse. We can use Pascal's theorem to find the line tangent to the ellipse at $A$: if $F$ is the intersection of $AB$ with $CD$, and $G$ is the intersection of $AC$ with $BE$, then the intersection $T_A$ of $FG$ with $DE$ is a point on the tangent at $A$ to the conic $ABCDE$ (see here for a detailed proof).
Draw through center $O$ the parallel $p$ to the tangent at $A$ and consider another point $B$ on the ellipse. Let $H$ be the intersection between line $OA$ and the parallel to $p$ through $B$, $K$ be the intersection between line $p$ and the parallel to $OA$ through $P$.
You can then construct point $A'$ on line $p$, such that $OA'$ is a conjugate semi-diameter of $OA$, finding $OA'$ from the equality (Apollonius equation for the ellipse): $$ \left({OK\over OA'}\right)^2+\left({OH\over OA}\right)^2=1. $$
Now that you have a couple of conjugate diameters you can construct the axes of the ellipse as explained here.
Finally, the three points you need are the reflections of the given point on the circle about the axes and about the center.