Wednesday, January 22, 2014

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

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