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All isogonal circum-strophoids with a node at I are nK with their root R on the trilinear polar of the Gergonne point X(7) containing the points X(241), X(514), X(650), X(665), X(905), X(1323), X(1375), X(1465), X(1638), X(3002), X(3004), X(3008), X(3015). See Special Isocubics §4.3.2. For non-isogonal circum-strophoids, see CL038. See also CL001, CL027 and a generalization in the page P-conical cubics.

Each cubic is the locus of foci of conics centered on a line L passing through the incenter I = X(1) and another point P = (p : q : r).

It is also spK(Z, X1) where Z is the infinite point of L, see CL055.

The singular focus F lies on the circumcircle of ABC since it is the isogonal conjugate of Z.

These strophoids form a pencil of circular cubics whose equation is :

(a^2 y z + b^2 z x + c^2 x y) p (c y - b z) = (x + y + z) b c p x (c y - b z),

where p (c y - b z) = 0 is the equation of the line passing through I and P,

and b c p x (c y - b z) = 0 is the equation of the circumconic which is the isogonal conjugate of the previous line i.e. the circumconic passing through I and P*.

The following table shows a selection of such remarkable strophoids.

line L

root R

focus F

infinity

cubic / other centers

remarks

IG

X(514)

X(106)

X(519)

K086 Gergonne strophoid

nodal tangents parallel to the asymptotes of the Feuerbach hyperbola

IO

X(905)

X(104)

X(517)

X(1339)

 

IH

R(515)

X(102)

X(515)

X(2077)

 

IX(5)

X(10015)

X(953)

X(952)

K165 NPC strophoid

 

IK

X(650)

X(105)

X(518)

K040 Pelletier strophoid

nodal tangents parallel to the asymptotes of the rectangular hyperbola with center X(116)

IX(7)

X(7658)

X(103)

X(516)

 

 

IX(21)

R(758)

X(759)

X(758)

 

nodal tangents parallel to the asymptotes of the Jerabek hyperbola

IX(39)

X(665)

?

?

K359 Brocard strophoid

 

IX(88)

X(3960)

?

X(2802)

X(2718)

 

IX(142)

X(3676)

X(1477)

?

X(2078)

 

IX(190)

X(4763)

X(2382)

?

X(537)

 

IX(474)

X(3669)

?

X(3880)

X(1319), X(1320)

 

IX(521)

X(1465)

X(108)

X(521)

 

 

IX(522)

X(3911)

X(109)

X(522)

 

 

IX(525)

X(1375)

X(112)

X(525)

 

 

IX(528)

X(1638)

X(840)

X(528)

X(7), X(55)

 

IX(740)

X(4369)

X(741)

X(740)

 

nodal tangents parallel to the asymptotes of the Kiepert hyperbola

IX(905)

X(241)

X(934)

X(3900)

 

 

 

 

 

 

 

 

Coordinates :

R(515) = (b - c)( -a^2 so + b^2 sc + c^2 sb + abc) : :

R(758) = a(b - c)(2 SA + bc) : :