The inner Napoleon circle, a term coined here for the first time, is the circumcircle of the inner Napoleon triangle. It has center at the triangle centroid (and is thus concentric with the outer Napoleon circle) and radius
(1)

where is the area of the reference triangle.
It has circle function
(2)
 
(3)

where and are Conway triangle notation. This function corresponds to the first isodynamic point , which is Kimberling center .
The only Kimberling center lying on it is , the first Fermat point.
The following table gives pairs of inverse Kimberling centers with respect to the inner Napoleon circle.
center  name  inverse center  name 
second Fermat point  first isodynamic point  
anticomplement of  complement of  
anticomplement of  complement of  
complement of  anticomplement of  
intercept of Euler line and line  fifth Moses intersection 