Areal centers

Areal center:

Let ΔA’B’C’ and ΔA”B”C” be two triangles inscribed in ΔABC. The areal center of ΔA’B’C’ and ΔA”B”C” is the point S such that the areas of ΔSA’A”, ΔSB’B” and ΔSC’C” are the same.

The next table shows the trilinear center function of the areal centers for every pair of named inscribed triangles.

Triangles		Areal center
EXTOUCH	INCENTRAL	X(1018)
EXTOUCH	INTOUCH	X(100)
EXTOUCH	LEMOINE	(4*a^2+b^2+c^2)*(a-b)*(a-c)/(4*a^2+b^2+c^2+3*a*(b+c)) On line: (662, 4069) ETC-search 6-9-13 = 2.279224265191568599975
EXTOUCH	MACBEATH	(a-c)*(a-b)*(S^2+SB*SC)/((-a+b+c)*(SB*b+SC*c+2*S*R)) On lines: (162,4551),(1618,2222) 0.04918013762886010149472
EXTOUCH	MEDIAL	X(100)
EXTOUCH	ORTHIC	X(101)
EXTOUCH	STEINER	(b^2-c^2)/(2*a^3-(b^2+c^2)*a-(b+c)*(b-c)^2) On lines: (1,4041), (65,4705), (75,4086) -3.910604001388742886439
EXTOUCH	SYMMEDIAL	(b^2+c^2)*(a-b)*(a-c)/(a*(b+c)+b^2+c^2) On lines: (163,643), (2298,4284) 4.063394297022946848274
EXTOUCH	YFF CONTACT	(b-c)/(2*a^2-a*(b+c)-(b-c)^2) On lines: (1,650), (2,522), (57,513), (81,3737), (88,1156), (89,4724), (105,1024), (279,3676), (676,2006), (885,3911), (1002,4893), (1022,3675), (1026,5376),  (1121,3227), (1638,2826), (3887,4845) -2.346744861647027810077
INCENTRAL	INTOUCH	X(4551)
INCENTRAL	LEMOINE	(a-b)*(a-c)*(b+c)*(4*a^2+b^2+c^2)/(a*(2*a^2-b^2-3*b*c-c^2)) On lines: (190,3908), (1723,5612) -0.4731678066924118032593
INCENTRAL	MACBEATH	(a-b)*(a-c)*(b+c)*(S^2+SB*SC)/(a*((b*c+SW)*SA-S^2)) On line: (692,1897) -0.1848021007747031358220
INCENTRAL	MEDIAL	X(3952)
INCENTRAL	ORTHIC	X(4559)
INCENTRAL	STEINER	(b^2-c^2)/(a*(a^2-b*c)) On lines: (10,514), (80,291), (313,3261), (513,894), (523,594), (741,2372),  (813,2690), (826,1089), (4013,4049), (4589,5380) 0.1903195606998795675211
INCENTRAL	SYMMEDIAL	(a-b)*(a-c)*(b+c)*(b^2+c^2) On lines:  (11,594), (100,101), (661,3952), (765,5389), (799,4562),  (1978,3807) 5.999448746196107405213
INCENTRAL	YFF CONTACT	(b^2-c^2)/(a*(a*(b+c)-2*b*c)) On Kiepert hyperbola On lines: (2,513), (10,661), (76,693), (98,739), (226,4017), (321,523),  (649,4672), (671,2787), (876,4358), (898,1290), (1647,4444), (2051,3667), (4010,4080) 0.7387428613738780332255+C2
INTOUCH	LEMOINE	(a-b)*(a-c)*(4*a^2+b^2+c^2)/(4*a^2-3*(b+c)*a+b^2+c^2) On line: (643,1019) 7.250456452563039901679
INTOUCH	MACBEATH	(a-b)*(a-c)*(SA^2+SB*SC)/(b*SB+c*SC-2*S*R) On line: (163,1021) -0.2097107384364576118994
INTOUCH	MEDIAL	X(100)
INTOUCH	ORTHIC	X(109)
INTOUCH	STEINER	(b^2-c^2)/(2*a^3-(b^2+c^2)*a+(b+c)*(b-c)^2) On lines: (9,661), (210,4705), (312,1577), (2321,4024), (2341,2610) 2.120399798139991690641
INTOUCH	SYMMEDIAL	(a-b)*(a-c)*(b^2+c^2)/(a*(b+c)-(b^2+c^2)) On lines: (101,4040), (662,3737), (666,1026), (831,919), (1581,2664) -2.263665312530479505429
INTOUCH	YFF CONTACT	(b-c)/(2*a^2-(b+c)*a+(b-c)^2) On Feuerbach hyperbola On lines: (1,3309), (7,3667), (8,514), (9,513), (21,1019), (80,2826), (104,1477), (294,1027), (314,5214), (764,3680), (876,4876), (900,3254), (1022,1280), (1023,1308), (1025,5377), (1156,2827) 2.308417730794870076935
LEMOINE	MACBEATH	(5*SW-3*SA)*(S^2+SB*SC)*(SA-SB)*(SA-SC)/(a*(SA^2+S^2-4*SB*SC)) -0.1732939223482771784293
LEMOINE	MEDIAL	(a^2-b^2)*(a^2-c^2)*(4*a^2+b^2+c^2)/a On lines: (2,353), (99,110), (111,5182), (669,1634), (892,4577), (1383,1992) 1.948852416877363244776
LEMOINE	ORTHIC	a*(a^2-b^2)*(a^2-c^2)*(4*a^2+b^2+c^2) On lines: (101,3518), (691,827), (1625,3050) 0.4513117194169890968045
LEMOINE	STEINER	(5*SW-3*SA)*(SB-SC)/(a*(2*SW^2-3*SW*SA-3*S^2)) On lines: (523,599), (598,1499) 2.036872078295437158369
LEMOINE	SYMMEDIAL	(b^2+c^2)*(a^2-b^2)*(a^2-c^2)*(b^2+4*a^2+c^2)/(a*(b^2+c^2-2*a^2)) On lines: (826,4576), (892,5466) -2.254331264696480276624
LEMOINE	YFF CONTACT	(3*SA-5*SW)*(b-c)/(a*((3*SA-5*SW)*a+(3*SB+SW)*b+(3*SC+SW)*c)) 2.243404035004790802826
MACBEATH	MEDIAL	(S^2+SB*SC)*(SA-SB)*(SA-SC)/(a*SA) Anticomplement of X(2972) On Johnson circumconic On lines: (2,1972), (4,94), (20,1075), (51,324), (107,110), (112,925), (162,655), (250,476),  (323,450), (436,1994), (523,1624), (653,3658), (685,4630), (877,4576), (1301,1302), (1897,4246), (2052,3060), (3186,4232) -0.07059630044964227674098
MACBEATH	ORTHIC	a*SA*(S^2+SB*SC)*(SA-SB)*(SA-SC) On Johnson circumconic On lines: (3,125), (25,114), (99,107), (110,351), (343,418), (476,930), (1625,2081), (3265,4576) -0.2649080012809499161772
MACBEATH	STEINER	(S^2+SB*SC)*(SB-SC)/(a*(SA+SW-6*R^2)) On line (6,2501) 1.734100567063112800206
MACBEATH	SYMMEDIAL	(SA-SB)*(SA-SC)*(SA+SW)*(S^2+SB*SC)/(a*(SA^2-SB*SC)) On line (879,2966) -0.2984878140101378863865
MACBEATH	YFF CONTACT	(S^2+SB*SC)*(b-c)/(((S^2+SB*SC)*a-(S^2-SA*SC)*b-(S^2-SA*SB)*c)*a) 1.474964369149482119831
MEDIAL	ORTHIC	X(110)
MEDIAL	STEINER	X(5466)
MEDIAL	SYMMEDIAL	X(4576)
MEDIAL	YFF CONTACT	(b-c)/(a*(b+c-2*a)) On circumhyperbola dual of Yff parabola On lines: (2,514), (7,3676), (75,693), (86,4833), (88,673), (106,675), (335,4080), (900,903),  (901,927), (918,4945), (1268,3004), (1647,4089), (1797,2989), (4025,4373), (4555,4618), (4615,5468)  -0.4590961383710460421040
ORTHIC	STEINER	a*(SB-SC)/((SA-SB)*SB+(SA-SC)*SC) Isogonal conjugate of X(4226) On Jerabek hyperbola On lines: (3,512), (4,3566), (6,924), (68,525), (69,523), (71,4079), (72,4705), (74,3563), (248,2422), (265,690), (526,895), (826,3519), (868,879), (1176,1510), (1499,4846) 10.35662876830228477958
ORTHIC	SYMMEDIAL	a*(SA+SW)*(SA-SB)*(SA-SC)/SA Isogonal conjugate of X(4580) On lines: (6,67), (25,694), (110,112), (162,660), (394,3162), (427,3051), (468,2211), (647,1624), (648,670), (933,2623), (1112,3124), (1576,2445) -0.4198495218937103159533
ORTHIC	YFF CONTACT	a*(b-c)/((SB+SC)*a-SB*b-SC*c) On lines: (3,649), (63,513), (295,3572), (1790,3733) 7.711263108722297722455
STEINER	SYMMEDIAL	(b^4-c^4)/(a*(a^2*(b^2+c^2)-2*b^2*c^2)) On lines: (2,512), (141,3005), (850,1502), (882,3266) 1.355867776715018247859
STEINER	YFF CONTACT	X(3120)
SYMMEDIAL	YFF CONTACT	(SW+SA)*(b-c)/(a*((2*SA+SB+SC)*a-(SA+SB)*b-(SA+SC)*c)) On lines: (86,3253), (334,876), (1930,2530) 1.672803830975516369642