Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least one pair of non-intersecting circles coplanar with the points, such that two of the given points are diameter endpoints of one circle, and another two of the given points are diameter endpoints of the other circle.
Tangent circles are considered to be non-intersecting. Coincident circles are considered to be intersecting.
I cannot find a counter-example, nor can I prove the conjecture. I made a generator of five (pseudo)random points.
(If, instead, we were given four points, the conjecture would not be true: for example, if the four points were the vertices of an equilateral triangle plus the centre, then we could not draw a pair of non-intersecting circles.)
Context: I was thinking about this question about random points in a disk. Staring at various sets of five points, I came up with this conjecture.
Edit: The generalized conjecture ($2n+1$ points and $n$ circles) is not true.
Edit 2: I have asked, and answered, a similar question. Maybe it might provide ideas for this question.
Edit 3 Posted on MO.