Dear Alex,
If S' = midpoint of XA, T' = midpont of YC
and as your notations in applet:
S = midpoint of AB
T = midpoint of BC
Q = midpoint of AC
Suppose
V = intersection of S'Q and SP
W = intersection of T'Q and TN
then S'Q//XC and SP//AY therefore angle(PVQ) = 60. It means V, B, Q, P are concyclic.
Similarly with W then we have more two concyclic points V and W.
Now we have total ten concyclic points.
Best regards,
Bui Quang Tuan