Consider a triangle ABC and a point D. For each of the triangles ABD, BCD, ACD construct the inscribed and circumscribed circles.
Conjecture: For any triangle ABC there always exists a special location of D, so that six intersection points of these six circles are concyclic.
Is this actually true or not?
This construction gives us 12 intersection points , but we are only interested in the "orange" ones and not in the "green" ones. In other words, we only consider the first intersection points of the arches ADB,BDA,BDC,CDB,ADC,CDA with the inscribed circles.
Comment:
This can be true. For equilateral triangle the center of this circle in on incenter (I)of the triangle because of symmetric quantity of vertices. In isosceles triangle the center of this circle is on the bisector of $\angle BAC$ as shown in figure (a). In other types the center of this circle has a distance from I, and it is not on any of the bisector of angles A, B and C as shown in figure (b).