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	*