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The real foci of the Brocard inellipse are the Brocard points Ω1, Ω2. They are a bicentric pair, with barycentrics 1/b^2 : 1/c^2 : 1/a^2 and 1/c^2 : 1/b^2 : 1/a^2, denoted by PU(1) in ETC, see Bicentric Pairs of Points.

In contrast to the real foci, the imaginary foci X(39641), X(39642) of the Brocard inellipse are major triangle centers. They both lie on the Brocard axis OK, on the Kiepert circumhyperbola, and on the curves in the table below.

They are obviously isogonal conjugates in ABC and symmetric in X(39). They are also inverse in the Brocard circle, and isogonal conjugates in the circumcevian triangle of K.

X(39641) = a^2(S + i SA) : b^2(S + i SB) : c^2(S + i SC) and X(39642) = a^2(S - i SA) : b^2(S - i SB) : c^2(S - i SC), with S = 2 area(ABC).

In KHO-coordinates, they are {√3, 0, i} and {-√3, 0, i}, see CL075.

Notes

• the cubics in the yellow cells are those of Table 51. They are the McCay stelloids spK(X3,Q). Their isogonal transforms are CircumNormal cubics, see Table 25.

• the cubics in the green cells are focal nKs, loci of foci of inconics with center on a line passing through X(39), the center of the Brocard inellipse. Every cubic passes through Ω1, Ω2. See other cubics below.

• the cubics in the orange cells are pKs. See further properties below.

pKs passing through X(39641) and X(39642)

Since X(39641), X(39642) are two isogonal conjugates on the Brocard axis, it is obvious that every isogonal pK with pivot P on the Brocard axis must contain these points. These cubics are in a same pencil with base-points A, B, C, the in/excenters and X(39641), X(39642).

Examples : K003 = pK(X6, X3), K102 = pK(X6, X6), K326 = pK(X6, X39), K763 = pK(X6, X1350).

Now, for any point M on the Brocard axis, pK(X54 x M, X54) also passes through X(39641), X(39642). The barycentric product X54 x M lies on the trilinear polar of X(14586) = X(54) x X(110), a line (L) passing through {160, 184, 571, 3135, 11135, 11136, 14533, 14573, 39231, 41270}.

Example : K373 = pK(14533, X54).

More generally, for any Ω ≠ X(6) and P ≠ X(54), there is one and only one pK(Ω, P) that passes through X(39641), X(39642).

Example : K049 = pK(X53, X4).

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With Ω = p:q:r and P = u:v:w, the transformations ΩtoP and PtoΩ are reciprocal and quadratic given by :

ΩtoP : p:q:r → b^2*c^2*(b^2 - c^2)*(b^2*c^2*p^2 - a^4*q*r) + a^4*p*(4*SA^2 - b^2*c^2)*(c^2*q - b^2*r) : : ,

PtoΩ : u:v:w → b^2*c^2*(b^2 - c^2)*(a^2*(b^2 + c^2) - (b^2 - c^2)^2)*u^2 + a^4*b^2*c^2*(b^2 - c^2)*v*w - a^2*u*(c^4*(4*SC^2 - a^2*b^2)*v - b^4*(4*SB^2 - a^2*c^2)*w) : : .

ΩtoP has three singular points X6 and two imaginary points Y1 and Y2, with barycentric product X(32), that lie on the line (L) above and on the rectangular circum-hyperbola (H5) passing through X(5). Every point on (L) is sent to X(54) and every point on (H5) is sent to a point on the Kiepert hyperbola.

Y1 is the intersection of the lines {X39641,X32046}, {X39642,X5} and Y2 is the intersection of the lines {X39642,X32046}, {X39641,X5}. Moreover, Y1 and Y2 are harmonically conjugated with respect to {X184,X571}.

The singular points of PtoΩ are X(54), X(39641), X(39642). Every point on the Brocard axis is sent to X(6).

Note that, for every point P on the Kiepert hyperbola, Ω = P x X(5) lies on (H5), hence pK(Ω, P) has its isopivot at X(5) and the tangents at A, B, C, P pass through X(5). This is the case of the stelloid K049 and pK(X5, X2).

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Y1 and Y2 lie on every pK(X32, P) where P is a point on the line (L). They also lie on the nodal cubic K1284 = cK(#X6, X1576).

Y1 and Y2 are now X(51539), X(51540) in ETC. They are the barycentric squares of X(39642), X(39641) respectively.

Circum-cubics passing through Ω1, Ω2, X(39641) and X(39642)

The circum-cubics passing through the four foci of the Brocard inellipse are K(P) = spK(P, X39) as in CL055, for some point P.

K(P) passes through A, B, C, Ω1, Ω2, X(39641), X(39642) which do not depend of P, and :

• P*, isogonal conjugate of P,

• P', reflection of P in X(39),

• the infinite points of pK(X6, P),

• the points of pK(X6, P') on the circumcircle (O),

• a third point P3 on the line {Ω1, Ω2}, which is the barycentric product of X(694) and the infinite point of the trilinear polar of the X(99)-isoconjugate of P,

• a third point Q3 on the Brocard axis {X3, X6}, which is the barycentric product of X(110) and the infinite point of the trilinear polar of the X(1916)-isoconjugate of P.

All these cubics are in a same net which can be generated by three decomposed independent cubics (Ka), (Kb), Kc), where (Ka) is the union of the line BC and the rectangular hyperbola passing through A, Ω1, Ω2, X(39641), X(39642).

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Special cubics of the net

• K516 = spK(X3, X39), the only stelloid of the net.

• with P on (L∞), K(P) is a focal nK(X6, R, P) with root R on the Lemoine axis (the trilinear polar of X6), with singular focus P* on (O). See K019, K166, K248, K359, K688 in the green cells of the table above.

• with P on the Brocard axis, K(P) is a K0 (without term in x y z). See K019, K166, K326, K516.

• with P on the circum-conic with center X(39) and perspector X(141), K(P) is a nK with pole Ω on nK(X32, X32, X6) passing through X(6), X(879), X(882) and root R on the circum-conic with perspector X(39) passing through X(67), X(110), X(660), X(670), X(694), X(1634).

• with P on the central cubic pK(X141, X194) with center X(39), K(P) is a psK with pseudo-pole on pK(X32, X32) and pseudo-pivot on K368 = pK(X3051, X6). This central cubic passes through X(39), X(76), X(194) – giving the cubics K326, K444, K512 – and also X(39468), X(42486), X(42551).

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Cubics passing though a given point M

K(P) passes through M if P lies on the line passing through M* and M' as above. For example :

• K(P) passes through X(6) if P lies on the trilinear polar of X(18829), passing through {2, 694, 1916, 2023, 3094, 3124, 3981, 4074, 4576, 5969, 5976, 6034, 8041, 8782, 8842, 9998, 10007, 11654, 13518, 18906, 20027, 20859, 24256, 25047, 25332, 27805, 32010, 33798, 34999, 36790, 36800, 39080, 40332, 40708, 44534, 45672, 46236, 47642, 47734, 48444, 49792, 50640, 52207, 52651, 56696, 56977, 57257, 59563, 59802}.

• K(P) passes through X(39) if P lies on the trilinear polar of X(41209), passing through {39, 83, 99, 384, 733, 1916, 3329, 5149, 7783, 7804, 7816, 8290, 9468, 10341, 10352, 14970, 32476, 38382, 40432, 51827, 51906, 55085, 56789}.

Remarks :

• Recall that X(6) and X(39) are the perspector and center of the Brocard inellipse.

• K(X99) decomposes into the Kiepert hyperbola and the line {Ω1, Ω2}, passing though {39, 512, 881, 882, 2524, 2531, 10567, 10568, 14251, 14824, 16068, 17415, 18872, 23099, 32484, 34347, 39680, 45882, 45887, 45900, 45901, 45902, 45903, 45904, 45905, 45906, 45907, 45908, 45909, 45910, 45911, 45912, 45913, 45914, 45915, 47648, 51494, 51511, 52700, 56748, 57453, 58262, 58779}.

• K(X1916) decomposes into the Brocard axis and the circum-conic with perspector X(385) passing through X(83), X(99), X(880).