Locus of P such that the CEVIAN 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
ANTICOMPLEMENTARY	The plane of ABC
BCI	cos(B/2)*(1+2*cos(A/2))*(cos(B/2)+1)*a*c^2 ETC: 174, 483, 1127	*
BROCARD1	a^2*(c*a-b^2)*(c*a+b^2)	K322
BROCARD2	c^2*(-b^4+c^4+a^4-c^2*a^2)	K531
BROCARD3	c^6*a^2*(c^2*a^2-b^4)	K532
BROCARD4	(c^2+a^2-b^2)*(-b^4+c^4+a^4-c^2*a^2)	K533
CIRCUMMEDIAL	b^2*(c^2+a^2) ETC: 2, 76, 83, 264, 308, 1799	*
CIRCUMORTHIC	SAME THAN CIRCUMMEDIAL	*
CIRCUMPERP1	Line (2,7) È Circumconic ∑(a*y*z)=0
CIRCUMPERP2	c*(a+c)	K317
COSYMMEDIAN	Cevians of X(6)
EULER	SA-2*S^2/SC ETC: 2, 4, 253, 3091	*
EXCENTRAL	All plane of ABC
EXTANGENTS	c*(a+b)*(b+c-a)	K033
EXTOUCH	{ }
FEUERBACH	(a+c)*(a+b)^2*(-b-c+a)*(-b^2+a^2+c*a+c^2) ETC: 1, 5, 12, 3615	*
FUHRMANN	F=b*(a^2-b^2+c*a)*(-b^2+c^2+a^2-c*a) G=(b-c)*(c-a)*(a-b)*(a+b+c)^2 ETC: 2, 10, 75, 1737, 2166	*
GREBE INNER	c^2*(b^2-S) ETC: 2, 4, 6, 1271, 5491	*
GREBE OUTER	c^2*(b^2+S) 2, 4, 6, 493, 1270, 5490	*
HEXYL	c*(a+c)*(a*SA-S*r)	K344
INCENTRAL	{ }
INTANGENTS	Feuerbach hyperbola
INTOUCH	{ }
JOHNSON	SB*(SW-SB)*(S^2+SB*SA) ETC: 2, 4, 5, 264, 311, 324, 847	*
LEMOINE	{ }
LUCAS CENTRAL	a^2*c^4*(a^2+b^2-c^2)*(2*S+b^2) ETC: 3, 6, 371, 588	*
LUCAS TANGENTS	a^2*c^4*((2*(3*b^2+a^2-c^2))*S+b^2*(4*a^2+b^2)- (a^2-c^2)^2) ETC: 6, 371, 493, 1151	*
MACBEATH	{ }
MEDIAL	{ }
MIDHEIGHT	b^2+c^2-a^2	K007
MIXTILINEAR	a*(a-b+c)*(SA+SB)(S^2-(2*(SC+SA))*c*a) ETC: 1, 57, 1697	*
MORLEY1	cos(A/3)*a*c^2*(2*cos(B/3)-1)*(2*cos(B/3)+1) ETC: 356, 357, 3602, 3604, 5456	*
MORLEY2	cos(A/3+Pi/3)*sin(B/3-Pi/3)*sin(B/3)*a*c^2 ETC: 1136, 3276, 3602, 3603	*
MORLEY3	sin(A/3+Pi/6)*cos(B/6-Pi/6)*sin(B/3)*a*c^2 ETC: 1134, 3277, 3603, 3604	*
MORLEYADJ1	cos(C/3)^2*cos(A/3)*(1-4*cos(B/3)^2)*a*c^2 ETC: 356, 358, 3602, 5456	*
MORLEYADJ2	cos(A/3+Pi/3)*cos(B/3+Pi/6)*sin(B/3)*a*c^2 ETC: 1137, 3276, 3603	*
MORLEYADJ3	(-1+2*cos((B+Pi)/3)*cos(B/3))*(-3+4*cos((C+Pi)/3)*cos(C/3))*sin(A/3+Pi/6)*a*c^2 ETC: 1135, 3277, 3604	*
NAPOLEON INNER	(2*(b^2-a^2-2*c^2))*S -sqrt(3)*(-c^2*(b^2+a^2)+(b^2-a^2)^2)	K420a
NAPOLEON OUTER	(-2*(b^2-a^2-2*c^2))*S -sqrt(3)*(-c^2*(b^2+a^2)+(b^2-a^2)^2)	K420b
NEUBERG1	((a^2+c^2)^2-c^2*a^2)*(a^2*b^2-b^4+c^2*a^2-c^4) ETC: 2, 25, 98, 183, 385, 3407	*
NEUBERG2	(a^2*(b^2+c^2-a^2)+2*b^2*c^2)*(b^4-c^2*a^2) ETC: 2, 262, 325, 427, 1916, 3329, 4518	*
ORTHIC	{ }
REFLECTION	(b^2+c^2-a^2)*((c^2+a^2-b^2)^2+c^2*a^2)	K060
SHARYGIN1	c*(a^2+b*c)	K132
SHARYGIN2	c*(a^2-b*c)	K323
SQUARES INNER	(c^2+a^2-b^2)*(b^2+c^2-a^2-2*S)	K070b
SQUARES OUTER	(c^2+a^2-b^2)*(b^2+c^2+a^2-2*S)	K070a
STEINER	{ }
SYMMEDIAL	{ }
TANGENTIAL	The plane of ABC
VECTEN INNER	SAME THAN SQUARES INNER
VECTEN OUTER	SAME THAN SQUARES OUTER
YFF CENTRAL	cos(B/2)^3*cos(A/2)*(cos(A/2)+cos(B/2))*a*c^2 ETC: 7, 174, 177, 234, 2089, 2091	*
YFF TANGENTS	{ }
YIU	(-2*cos(B)-2*cos(3*B+2*C)*cos(2*C)+  2*cos(B)*cos(2*C)+cos(6*C+3*B)-2*sin(2*B+3*C)* sin(B-C))*sin(B)*sin(C)*cos(C+2*B) ETC: 5, 1994	*