Locus of P such that the ANTICEVIAN triangle of P and a given triangle are perspective

Most of loci are cubics, except where indicated. Barycentrics equations of the cubics are writen as: ∑ [ (F(a, b, c)*y2 - F(a, c, b)*z2 )*x ]  + G(a, b, c)*x*y*z  + ∑ [ H(a, b, c)*x3 ]= 0 In this table, F is given in first term, H=0 for all triangles and G=0 except for FUHRMANN triangle. Also, ETC indexes for points on the cubic are given. Gibert's catalogue of cubics of triangles: http://bernard.gibert.pagesperso-orange.fr

TRIANGLE	LOCUS OF P (F, G and ETC indexes)	Gibert's catalogue
ABC	The plane of ABC
ANTICOMPLEMENTARY	Cevians of X(2)
BCI	(cos(A/2)-cos(B/2)-cos(C/2)-2*cos(B/2)*cos(C/2))*a*c^2 ETC: Excenters and 1, 173, 258, 1127, 1129	*
BROCARD1	c^2*(-b^2*c^2+a^4)	K128
BROCARD2	c^2*(b^2+c^2-2*a^2)*(a^2+b^2-c^2)	K534
BROCARD3	c^2*(-b^2*c^2+a^4)	K128
BROCARD4	c^2*(b^2+c^2-2*a^2)*(a^2+c^2-b^2)	K535
CIRCUMMEDIAL	c^2*(c^2+a^2) ETC: 2, 4, 6, 83, 251, 1176, 1342, 1343	K644
CIRCUMORTHIC	(SA+SB)*(SC*SA+S^2)*SB ETC: 2, 4, 6, 54, 275, 1993	*
CIRCUMPERP1	Line(1, 6) È Circumconic ∑a^2*(-b-c+a)*y*z=0
CIRCUMPERP2	a*c^3*(a+c)	K319
COSYMMEDIAN	Cevians of X(6)
EULER	(SB*SA+S^2)*SB*SA ETC: 2, 4, 5, 53, 216, 1249, 2052	*
EXCENTRAL	Cevians of X(1)
EXTANGENTS	a*c^3*(a+b)	K362
EXTOUCH	The plane of ABC
FEUERBACH	a*(a^2-b^2)^2 ETC: 1, 11, 12, 523, 1109, 2588, 2589, 2616, 2618	*
FUHRMANN	F=c^2*(-a*b^2-b*a^2+c*a^2+a^3+b^3-b*c^2) G=(b-c)*(c-a)*(a-b)*(a+b+c)^2 ETC: 1, 2, 6, 106, 1465, 1718, 2006	*
GREBE INNER	c^2*(a^2+b^2-S) ETC: 2, 3, 6, 3128	*
GREBE OUTER	c^2*(a^2+b^2+S) ETC: 2, 3, 6, 3127	*
HEXYL	(b^2+c^2-a^2)*a*c^2	K343
INCENTRAL	The plane of ABC
INTANGENTS	Line (6,9) È Line (44,513)
INTOUCH	The plane of ABC
JOHNSON	SC*(SA+SB)*(S^2+SB*SA) ETC: 2, 3, 5, 6, 216, 343, 2165	*
LEMOINE	The plane of ABC
LUCAS CENTRAL	(a^2+b^2-c^2)*(a^2+b^2-c^2+4*S)*c^4*a^2 ETC: 3, 6, 3167, 3311, 5406	*
LUCAS TANGENTS	(a^2+b^2-c^2)*(a^2+b^2-c^2+2*S)*c^4*a^2 ETC: 3, 6, 371, 3167, 5408	*
MACBEATH	The plane of ABC
MEDIAL	The plane of ABC
MIDHEIGHT	(SA+SB)*(S^2-2*SB*SC)	K004
MIXTILINEAR	a*(a+b-c)*(a-b+c)*(SA+SB)*(-SA*a-SB*b+c*SC+2*S*R) ETC: 1, 40, 57, 1743, 2324	*
MORLEY1	a*c^2*(cos(A/3)-2*cos(C/3)*cos(B/3))	K585
MORLEY2	a*c^2*sin(C/3)*sin(B/3) ETC: Excenters and 1, 1136, 1137, 3273, 3275, 3603, 3604	*
MORLEY3	(sqrt(3)*cos(A/3-Pi/6)+2*sin(C/3)*sin(B/3))*a*c^2 ETC: Excenters and 1, 1134, 1135, 3273, 3274, 3602, 3604	*
MORLEYADJ1	Same than MORLEY1	K585
MORLEYADJ2	Same than MORLEY2	*
MORLEYADJ3	Same than MORLEY3	*
NAPOLEON INNER	(3*SW-3*SA-2*sqrt(3)*S)*(SA+SB)	K129a
NAPOLEON OUTER	(3*SW-3*SA+2*sqrt(3)*S)*(SA+SB)	K129b
NEUBERG1	c^2*(a^4-b^2*c^2)	K128
NEUBERG2	c^2*(2*c^2*a^2+a^4+b^2*c^2+2*a^2*b^2)	K423
ORTHIC	The plane of ABC
REFLECTION	c^2*(c^2*a^2+2*b^2*c^2-c^4-b^4+a^2*b^2)	K005
SHARYGIN1	a^2*c^2*(-c^2+a*b) ETC: 1, 6, 43, 81, 238, 239, 256, 291, 294, 1580, 2068, 2069, 2238, 2665	*
SHARYGIN2	SAME THAN SHARYGIN1
SQUARES INNER	c^2*(a^2+b^2-c^2)*(c^2+a^2-b^2)	K006
SQUARES OUTER	SAME THAN SQUARES INNER	K006
STEINER	The plane of ABC
SYMMEDIAL	The plane of ABC
TANGENTIAL	Cevians of X(6)
VECTEN INNER	(SA+SB)*(SA-SW+S)	K424b
VECTEN OUTER	(SA+SB)*(SA-SW-S)	K424a
YFF CENTRAL	(cos(A/2)+cos(B/2))*(cos(B/2)+cos(C/2)-cos(A/2)) *cos(B/2)*a*c^2* ETC: 1, 177, 188, 2089	*
YFF TANGENTS	The plane of ABC
YIU	c^2*a^2*(-b^2*c^2-c^2*a^2+b^4-2*a^2*b^2+a^4)*(b^4+c^4-b^2*c^2+a^4-2*a^2*b^2-2*c^2*a^2)*(a^4*b^2*c^2+2*a^4*b^4+b^4*a^2*c^2-2*b^6*a^2+5*b^2*a^2*c^4-2*a^6*b^2+a^8+b^8+c^8-4*c^2*a^6+6*c^4*a^4-4*c^6*a^2+6*b^4*c^4-4*b^6*c^2-4*b^2*c^6) ETC: 6, 195	*

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

Recursive functions with cyclic sums

RECURSIVE FUNCTIONS WITH CYCLIC SUMS

The recursive equalities shown here make easier working with cyclic sums applied to triangles, specially when a conversion of (a,b,c) to (R,r,s) is required, i.e, when some triangle data must be expressed as function of  R (circumradius), r (inradius) and s (semi-perimeter).

Let’s name

σ(n) = ∑ [ a­­^n­ ]

ρ(n) = ∑ [ (b c)­­^n­ ]

λ(k, m, n)=∑ [ a^k (b^m cⁿ + bⁿ c^m)­­^­ ]   m>=n

where k, n, m are integer numbers and all sums are cyclic, and

Π(n) = (a b c)ⁿ = [Π(1)]ⁿ

General expression for σ(n):

It is obvious that σ( 0 ) = 3 and σ( 1 ) = 2 s and it is well known that σ(2)=a² + b² + c²  = s² – r² – 4 R r and   Π(1)=4*R*r*s.

For other side, if n<0 then

σ(n) = ∑ [ 1/a­­^|n|­ ] = (a b c)ⁿ ∑ [ (b c)­­^|n|­ ] = Π(n) ρ(-n)       [n<0]

 Suppose now that n>=3 and let’s calculate σ(1) σ(n-1), i.e.

σ(1) σ(n-1) = ∑ [ a­­ ] ∑ [ a­­^n­-1 ]  =

σ(n) + a ( b^n-1 + c^n-1 ) + b ( c^n-1 + a^n-1 ) + c ( a^n-1 + b^n-1 ) =

σ(n) + b c ( b^n-2 + c^n-2 ) + c a ( c^n-2 + a^n-2 ) + a b ( a^n-2 + b^n-2 ) =

σ(n) + b c (σ(n-2) - a^n-2 ) + c a (σ(n-2) - b^n-2) + a b (σ(n-2) - c^n-2) =

σ(n) + σ(n-2) ∑ [ b c­^­ ] – a b c σ(n-3) =

σ(n) + ρ(1) σ(n-2) – Π(1) σ(n-3)

from which it is possible to deduce σ(n) as:

σ(n) = ∑ [ a­­n­ ] =

Π(n) ρ(-n)                                                                  if n<0 3                                                                                if n=0 2 s                                                                             if n=1 s2 – r2 – 4 R r                                                           if n=2 σ(1) σ(n-1) - ρ(1) σ(n-2) + Π(1) σ(n-3)                    if n>=3

In a similar way it is easy to obtain expressions for ρ(n) ( for n>0 just get it from (aⁿ + bⁿ +cⁿ)²  ) and (with a little more effort) for  λ(k, m, n). These are:

ρ(n) = ∑ [ (b c)­­n­ ] =

Π(n) σ(-n)                                         if n<0 3                                                       if n=0 ( σ(n)² - σ(2 n) )/2                            if n>0

and:

λ(k, m, n)=∑ [ a^k (b^m cⁿ + bⁿ c^m)­­^­ ]  = Π(n) ( σ(m-n) σ(k-n) - σ(k+m-2 n) )    

for m>=n 

Note that σ(n) and ρ(n) call each one the other. It can be checked that this fact does not lead to any infinite loop.

Also note that these functions can be extended to other cyclic sums: you only need their cyclic product and the cyclic sums of the powers 0, 1 and 2 (…and power 0 is always equal to 3). For example, for working with σ(n) = ∑ [cos(A) ]ⁿ, ρ(n) = ∑ [cos(B) cos(C) ]ⁿ and λ(k,m,n) = ∑ [cos^k(A) (…) ] you only need:

σ(1) = (R+r)/R,  σ(2) = (-s²+6R²+4Rr+r²)/(2*R²) and Π(1)= (s²-4R²-4Rr-r²)/(4*R²)

The implementation of these functions in Maple and a small procedure (which determines what function and parameters to use) gave me, in half-one-minute, the 48 combos visible by pressing the next button:
    

X(4) = 3*X(2)-2*X(3)

X(5) = 3*X(2)-X(3)

X(17) = (s^2+r^2+2*r*R)*X(1)+2*r*(2*sqrt(3)*s-3*R)*X(2)-(2*(sqrt(3)/3*s+r))*r*X(3)

X(18) = (s^2+r^2+2*r*R)*X(1)-2*r*(2*sqrt(3)*s+3*R)*X(2)+(2*(sqrt(3)/3*s-r))*r*X(3)

X(22)=3*R^2*X(2)+(-s^2+r^2+4*r*R)*X(3)

X(23)=9*R^2*X(2)+2*(-s^2+r^2+4*r*R)*X(3)

X(24)=3*R^2*X(2)+(4*r*R+r^2-s^2+2*R^2)*X(3)

X(25)=6*R^2*X(2)+(-s^2+r^2+4*r*R)*X(3)

X(26)=3*R^2*X(2)+(4*r*R+R^2-s^2+r^2)*X(3)

X(27)=3*(4*r*R+4*R^2+s^2+r^2)*X(2)-4*s^2*X(3)

X(28)=3*R*(r+2*R)*X(2) + (-s^2+r^2+2*r*R)*X(3)

X(29)=3*(4*R^2-s^2-r^2)*X(2)+4*r*(r+2*R)*X(3)

X(35)=R*X(1)+2*r*X(3)

X(36)=R*X(1)-2*r*X(3)

X(42)=-(s^2+r^2+4*r*R)*X(1)+6*r*R*X(2)

X(43)=-(s^2+r^2+4*r*R)*X(1)+12*r*R*X(2)

X(46)=(r+R)*X(1)-2*r*X(3)

X(53) = ((r+2*R)^2-s^2)*(s^2+r^2+2*R*r)*X(1)-6*r*(R*(r+2*R)^2-s^2*(R+2*r))*X(2)-2*r^2*((r+2*R)^2+3*s^2)*X(3)

X(54) = (s^2+r^2+2*R*r)*X(1)-3*R*(R+2*r)*X(2)+(-4*R*r-2*R^2+s^2-3*r^2)*X(3)

X(57)=(2*R+r)*X(1)2*r*X(3)

X(58) = (s^2+r^2+2*R*r)*X(1)-6*R*r*X(2)-4*r^2*X(3)

X(59)=(r^3-2*r*R^2+4*R^3+r*s^2+r^2*R-s^2*R)*X(1)+3*R*r*(R-2*r)*X(2)-r*(3*r^2+4*R*r+4*R^2-s^2)*X(3)

X(60) = (s^2+r^2+2*R*r)*(r+R)*X(1)-3*R*r*(3*R+2*r)*X(2)-r*(4*R*r-s^2+3*r^2)*X(3)

X(61) = (s^2+r^2+2*R*r)*X(1)-6*R*r*X(2)+(2*(sqrt(3)/3*s-r))*r*X(3)

X(62) = (s^2+r^2+2*R*r)*X(1)-6*R*r*X(2)-(2*(sqrt(3)/3*s+r))*r*X(3)

X(64)=(s^2+r^2+2*R*r)*X(1)-6*R*(r+2*R)*X(2)+2*(4*R*r+8*R^2-s^2)*X(3)

X(65) = (r+R)*X(1)-r*X(3)

X(66) = (2*R^2+4*R*r-s^2+r^2)*(s^2+r^2+2*R*r)*X(1)+3*(-s^2+r^2+2*R*r)*(2*R^2+4*R*r-s^2+r^2)*X(2)+(-20*R^2*r^2+8*R*r*s^2-16*r^3*R-s^4+4*s^2*r^2-3*r^4)*X(3)

X(67) = (3*(s^2+r^2+2*R*r))*(-s^2+r^2+4*R*r+3*R^2)*X(1)+(6*(-s^2+r^2+R*r))*(-s^2+r^2+4*R*r+3*R^2)*X(2)+(-34*R^2*r^2-s^4+8*s^2*r^2-7*r^4+8*R*r*s^2-32*r^3*R)*X(3)

X(68) = (s^2+r^2+2*R*r)*X(1)+3*(-s^2+r^2+2*R*r+2*R^2)*X(2)+(-2*R^2+s^2-3*r^2-4*R*r)*X(3)

X(70) = (s^2+r^2+2*R*r)*(4*R*r+3*R^2-s^2+r^2)*X(1)+(3*(r+R+s))*(r+R-s)*(4*R*r+3*R^2-s^2+r^2)*X(2)+(-s^4+4*s^2*r^2-3*r^4-16*r^3*R-25*R^2*r^2-12*r*R^3+8*R*r*s^2+3*R^2*s^2-2*R^4)*X(3)

X(71) = (4*R*r+s^2+r^2+4*R^2)*X(1)-6*R*(r+2*R)*X(2)+(-2*s^2-2*r^2-4*R*r)*X(3)

X(75) = 2*(s^2-r^2-2*R*r)*X(1)-3*(s^2+r^2)*X(2)+4*r^2*X(3)

X(76) = (2*(s^2+r^2+2*R*r))*(-s^2+r^2+4*R*r)*X(1)+(3*((s^2+r^2)^2+4*R*r*(-s^2+r^2)))*X(2)-4*r^2*(s^2+r^2+4*R*r)*X(3)

X(77) = (2*R*r+4*R^2-s^2)*X(1)+3*R*r*X(2)-4*R*r*X(3)

X(78) = (2*R+r)*X(1)-3*R*X(2)

X(79) = (R/2+r/3)*X(1)+r*X(2)-r*X(3)

X(82) = (6*R*r*s^2-4*R^2*r^2-s^4+r^4+2*r^3*R)*X(1)-3*r*R*(-s^2+3*r^2+8*R*r)*X(2)-4*r^2*(3*R*r-s^2+r^2)*X(3)

X(83) = (2*(s^2+r^2+2*R*r))*(-s^2+r^2+4*R*r)*X(1)+(-3*s^4-6*r^2*s^2+36*R*r*s^2-3*r^4-36*r^3*R-96*R^2*r^2)*X(2)-4*r^2*(-3*s^2+r^2+4*R*r)*X(3)

X(85) = (2*s^2-2*r^2-12*R*r-16*R^2)*X(1)+(-12*R*r-3*s^2-3*r^2)*X(2)+4*r*(r+4*R)*X(3)

X(87) = (8*R*r*(s^2+r^2+2*R*r)-(s^2+r^2)^2)*X(1)-12*R*r*(4*R*r-r^2-s^2)*X(2)-32*r^3*R*X(3)

X(89) = (4*R*r+(2/3)*s^2+2*r^2)*X(1)-3*R*r*X(2)-4*r^2*X(3)

X(91) = (R*(8*R*r+2*R^2-s^2)+r*(2*r^2+7*R*r-2*s^2))*X(1)-3*r*(2*R^2-s^2-r^2)*X(2)-4*r^2*(r+R)*X(3)

X(93) = (2*(s^2+r^2+2*R*r))*(r+2*R+s)*(r+2*R-s)*X(1)+(3*(s^4+5*R^4+4*r^3*R+2*r^2*s^2+8*r*R^3+r^4-4*R*r*s^2+6*R^2*r^2-6*R^2*s^2))*X(2)+(2*(-R^4-9*R^2*r^2-2*r^4+R^2*s^2-8*r^3*R-4*r*R^3-2*r^2*s^2))*X(3)

X(94) = (2*(s^2+r^2+2*R*r))*(3*R^2+4*R*r-s^2+r^2)*X(1)+3*(s^4+9*R^4-4*R*r*s^2+r^4+2*r^2*s^2+6*R^2*r^2-6*R^2*s^2+4*r^3*R+9*r*R^3)*X(2)-4*r^2*(s^2+3*R^2+r^2+4*R*r)*X(3)

X(95) = (2*(r+2*R-s))*(r+2*R+s)*(s^2+r^2+2*R*r)*X(1)+(12*R^2*r^2-12*R^2*s^2+6*r^2*s^2+3*r^4-12*R*r*s^2+3*s^4+12*r^3*R)*X(2)-4*r^2*(-3*s^2+4*R^2+4*R*r+r^2)*X(3)

X(96) = (2*(s^2+r^2+2*R*r))*(2*R^2-s^2+4*R*r+r^2)*X(1)+(12*R^4+12*R^2*r^2-12*R*r*s^2+3*s^4+12*r^3*R+3*r^4+6*r^2*s^2-12*R^2*s^2+24*r*R^3)*X(2)-(2*(2*R^2-s^2+4*R*r+r^2))*(-s^2+2*R^2+3*r^2+4*R*r)*X(3)

X(97) = -(s^2+r^2+2*R*r)*(r+2*R+s)*(r+2*R-s)*X(1)+3*R*(2*r+R)*(r+2*R+s)*(r+2*R-s)*X(2)+2*r^2*(-3*s^2+4*R^2+r^2+4*R*r)*X(3)

NOTE ADDED Sept. 18, 2013-21:35: Barry Wolk kindly informed me that Maple can do this same work just using "Simplify with side relations". I appreciate this tip very much, which I ignored. Thank you, Barry.

Conic through 40 ETC centers - Locus

The circumconic U with trilinear equation:

U : ∑ [ (b^2 – c^2)* v * w ] = 0

having center X(244) and perspector X(661) passes through A, B, C and these 40 ETC centers: X(1), X(10), X(19), X(37), X(65), X(75), X(82), X(91), X(158), X(225), X(267), X(596), X(759), X(775), X(876), X(897), X(921), X(969), X(994), X(1247), X(1581), X(1910), X(2153), X(2154), X(2166), X(2168), X(2186), X(2190), X(2214), X(2216), X(2217), X(2218), X(2219), X(2363), X(2588), X(2589), X(2652), X(2962), X(3668), X(4674).

U is the locus of P (not on the circumcircle of ABC) such that the antipedal triangle of P w/r to a ABC and the incentral triangle of ABC are orthologic.

X(759) is the fourth intersection of U and the circumcircle.

If O1(P) is the orthologic center (Incentral ; P-Antipedal ) for P then ( P, O1(P) )=( X(I), X(J) ) for these (I,J): (1,1), (10, 4065), (19,4319), (37, 2667), (65,2292), (75,192), (596,3159), (759,5497)

Some non-ETC O1(P) are:

P	O1(P)	Details of O1(P)
X(82)	(a^2+b^2+c^2-a*(b+c)+b*c)*(a^2+b^2-b*c+c^2)	On lines: (1,2896), (37,82), (192,3938), (744,2667), 3057,3100), (3159,3685), (4085,5262), (4514,4972)
X(876)	(a^2-b*c)*(b^2+c^2-a*(b+c))*(b-c)	Reflection of X(I) on X(J) for these (I,J): (2254,665), (3766,3716) On lines: (1,514), (37,513), (190,5378), (192,522), (512,2292), (523,2667), (649,4414), (659,4435), (665,1642), (764,4983), (891,3251), (1281,2785), (1742,3667), (3057,4083), (3716,3766), (4065,4151), (4124,4448)
X(897)	(a^2+b^2+c^2-3*b*c)*(a^2+b^2+c^2-3*a*(b+c)+3*b*c)	On lines: (1,2796), (37,100), (2836,3057), (4442,4956)
X(969)	(a^2+b^2+c^2+4*b*c)*(a^2+b^2+c^2+2*a*(b+c))	On lines: (1,193), (37,63), (192,612), (975,3159), (1773,3743), (2667,3870), (3920,4319), (4461,5297), (4657,5241)
X(994)	(a-2*(b+c))*(b^2+c^2+a*(b+c)-b*c)	On lines: (1,89), (2,4674), (8,3159), (37,517), (42,3899), (45,4752), (145,4065), (190,996), (192,519), (386,3878), (514,1000), (750,3245), (758,2667), (982,3898), (984,2802), (986,3884), (991,2800), (995,3877), (2099,4653), (2177,4867), (3670,3890), (3679,4125), (3727,3730), (3938,5497), (4256,5289)
X(2214)	(a^2+(b+c)^2)*(a^2+b^2+c^2+a*(b+c))	On lines: (1,69), (31,37), (192,3920), (344,1961), (534,4319), (612,2345), (1486,1962), (2667,3938), (3057,4336), (3159,3923), (3966,4657)

The locus of O1 is the conic V1 with trilinear equation:

V1: ∑ [ a*(b-c)*(u^2 + v*w) ] = 0

The center of V1 is:

   (b+c)*(a*(b+c)*(a^2-b*c)+(b^2-4*b*c+c^2)*a^2+2*b^2*c^2) : :

and lies on lines: (1,4427), (42,3952), (244,1962), (659,3722), (740,899), (1193,4065), (2292,2611), (2667,4117), (2802,3743)

V1 passes through the vertex of the incentral triangle and these 10 ETC centers: X(1), X(37), X(192), X(2292), X(2667), X(3057), X(3159), X(4065), X(4319), X(5497) .

Some non-ETC P (given O1(P)):

O1(P)	P	Details of P
X(37)	1/(a^2-a*(b+c)-b*c)	Isogonal conjugate of X(1621)  Midpoint of (3555,3696)  On lines: (10,141), (37,38), (65,1418), (75,3873), (81,82), (225,1876), (244,872), (596,740), (674,3664), (692,3449), (876,4132), (1002,4000), (1037,5228), (1468,2218), 1486,3423), (3286,3941), (3446,5091), (3555,3696), (3668,5173), 3681,4751), (3742,4698), (3779,4675), (4032,5083), (4430,4699)
X(3057)	1/(a^3-(b^2-b*c+c^2)*a-b*c*(b+c))	Isogonal conjugate of X(2975) On lines: (1,859), (5,10), (28,2190), (31,2217), (37,1953), (48,2214), (60,1610), (65,1193), (72,4692), (75,3869), (143,952), (197,1036), (214,5482), (225,1829), (595,759), (596,758), (957,3086), (961,3450), (1460,3435), (2218,3915), (2390,4292), (2933,5264), (3216,4674), (3668,3827)

The orthologic centers O2(P) (P-Antipedal ; Incentral) have complicated expressions. The only ETC-defined correspondence is O2( X(1) ) = X(3). The locus of O2(P) is also a conic with center of few interest.