The minimum number of determined circles is the least number 
of distinct circles through
at least three points of an 
-point set in the plane
whose points are neither all collinear nor all concyclic,
for 
(Elliott 1967). A circle through more than three of the
points is counted once. Erdős (1961) posed the question with the concyclic
case excluded. Elliott (1967) also excluded the collinear
case, which would give zero circles. This is problem 506
in the Erdős problems.
The minimum is
 |
(1)
|
Here 
denotes the floor function. For 
, 5, or 
,
a minimizing configuration consists of the center of a
circle together with 
points on it. The points on the circle
form 
antipodal pairs and, when 
is even, a single unpaired point. Each nonantipodal pair determines
a distinct circle through the center,
in addition to the original circle (Purdy and Smith 2010,
Cambie et al. 2026).
Jucovič (1967) determined the cases from 4 through 6, and Purdy and Smith (2010) proved the formula for 
.
Wang (2026) gave the complete classification. Cambie et al. (2026) gave shorter
proofs for 
and 
.
The illustrated extremal eight-point configuration has four points 
for 
, 
, 
,
and 
, where 
.
The other four points are 
and 
.
These eight points determine 11 circles through four points
each and six ordinary circles, each containing exactly
three of the points (Cambie et al. 2026).
Points." Acta Math. Acad. Sci. Hungar. 18,
181-188, 1967.