Concurrent perpendicular bisectors.

Let ΔA’B’C’ be a triangle T of a triangle ΔABC. The parallel line to (AB) through B’ cuts (BC) in Ab and the parallel line to (AB) through C’ cuts (BC) in Ac. Build Ba, Bc, Ca, Cb cyclically.  The perpendicular bisectors of [AbAc], [BaBc] and [Ca,Cb] concur at a point Z(T)  for some triangles T.

The next table shows Z(T) for such triangles. 

TRIANGLE T of ΔABC	Z(T): Trilinear center function and properties	ETC-(6,9,13)-search
ANTICOMPLEMENTARY BROCARD1 NAPOLEON INNER NAPOLEON OUTER NEUBERG1 NEUBERG2 VECTEN INNER VECTEN OUTER	X(3)	--
BROCARD2	a*(18*S^2*SA^2-3*SW*(SW^2+9*S^2)*SA+S^2*(9*S^2-SW^2)) On lines: (2,1495), (353,511), (575,1383)	8.96309921686445
COSYMMEDIAN	a*(18*S^2*SA^2-(54*S^2*R^2+SW*(9*S^2+SW^2))*SA+S^2*(9*S^2-SW^2)) On line: (182,381)	11.14383553953250
GREBE INNER	a*((S-SW)*SA+S^2) On Brocard Axis (3-6) Midpoint of: (3,1161) On lines: (3,6), (4,487), (25,5409), (30,1991), (51,1584), (325,637), (394,3155), (1306,3563), (1583,3917), (1599,2979), (1600,3060), (3156,5407)	-10.71004782008930
GREBE OUTER	a*((S+SW)*SA-S^2) On Brocard Axis (3-6) Midpoint of: (3,1160) On lines: (3,6), (4,488), (25,5408), (51,1583), (325,638), (394,3156), (1307,3563), (1584,3917), (1599,3060), (1600,2979), (3155,5406)	24.27477360848200
SYMMEDIAL	a*(2*S^2*SA^2-(SW*(S^2+SW^2)+8*S^2*R^2)*SA+S^2*(S^2-SW^2)) On lines: (3,695), (4,83)	4.67367749959897
TANGENTIAL	a*(2*SA^2-(SW+4*R^2)*SA+S^2) Midpoint of: (3,1498), (1352,5596) Reflection of: (182/206), (1147/156), (3357/3) On lines: (3,64), (4,54), (5,182), (6,1598), (20,110), (22,5562), (24,185), (25,389), (30,156), (32,1971), (49,382), (52,161), (68,542), (107,1075), (155,159), (186,1204), (221,999), (381,569), (399,2917), (567,3843), (576,2393), (1012,1437), (1216,3098), (1352,3547), (1385,5248), (1397,3073), (1656,1853), (1872,2182), (1899,3542), (1968,3331), (1974,3089), (2175,3072), (2192,3295), (2781,5609), (2818,3556), (3091,5012)	-5.64011769172535