Saturday, March 15, 2014

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
*
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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
*


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