In the plane there are six different points , , , , , such that and are parallelograms. What is the maximum number of those points that can be located on one circle?
Answer: 5.
Solution
As and are parallelograms, the line segments , and are parallel and have same length. Since it is impossible to draw three chords of equal length to a circle, not all 6 points can be concyclic.
Figure 3
Figure 4
A construction with 5 concyclic points is in fig. 3.
Note. There are many constructions with 5 vertices. We can, e.g., take a rectangle , add the fifth point randomly on the circumcircle of the rectangle and choose point such that would be a parallelogram (see fig. 4).
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