Given any three distinct points $A,B,C$ and a circle $C(O)$, construct points $D,E,F$ on the circle such that
- $A,D,E$ are collinear,
- $B,E,F$ are collinear and,
- $C,F,D$ are collinear.
One such solution is indicated on the diagram below. I have enough analytic and numerical evidence to indicate that these points exists. In fact, there are two such sets of points, as shown by @coproc below. However I would like a geometric construction.
One idea: My idea was to invert the points $A,B,C$ in the circle to find points $A',B',C'$ respectively. The problems then becomes equivalent to the following problem.
Given three points $A',B',C'$ and a circle $C(O)$. Construct three circles such that
- One circle passes through points $A'$ and $O$,
- another passes through points $B'$ and $O$,
- the third passes through points $C'$ and $O$ and,
- the three circles intersect pairwise on $C(O)$.
The required points $D,E,F$ are just the points of intersection of these three circles.