Here is a construction I came upon recently:
A'B'C' is the contact triangle. X(1) is the Incenter. A'',B'',C'' are the midpoints of the sides of the contact triangle. A''',B''',C''' are the midpoints of the segments AX(1), BX(1), CX(1). Then quadrilaterals A''A'''C''C''', B''B'''C''C''', A''A'''B''B''' are cyclic. Let Ob, Oa, Oc be the circumcenters of these 3 quadrilaterals. Finally by drawing circles with centers at Oa, Ob, Oc and passing through the incenter we get the 'inscribed-circumscribed' six point circle with its center at the midpoint of X(1) and X(3) - the point X(1385) in the ETC:
There is an even easier way to get a smaller concentric six point circle:
circumcircles of the quadrilaterals A''A'''C''C''', B''B'''C''C''', A''A'''B''B''' cut the sides of the triangle ABC at six concyclic points.
The mere fact that the point X(1385) is already described in the ETC supposedly makes the construction not particularly inspiring for a geometer. However as this 'X(1385)-circle' has never been previously mentioned (?) in the literature, there might still be a chance that it has some 'nice' properties of its own.
- Specifically I am interested whether any Kimberling center belongs to this 'inscribed-circumscribed' circle? Or, alternatively, does its central function correspond to any known center?