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A contribution by Wilson Stothers (8 November 1946  16 July 2009) This page is dedicated to his memory *** Preamble : extraversion, strong and weak geometric objects On this page, we use the algebraic form of extraversion as in Hyacinthos #7250. For a point X, {X} is the set of algebraic conjugates of X over the field generated by a^2, b^2, c^2. Then {X} is the set of algebraic extraversions of X. A function f(a,b,c) is a strong function if it is rational in a^2, b^2, c^2. An algebraic curve is a strong curve if it has an equation whose coefficients are strong functions. A point is a strong point if it has barycentric coordinates which are strong functions. Otherwise, X is a weak point. Equivalently, X is a strong point if the set {X} consists of X itself. Thus, X2, X3, X4, X5, X6 are strong, while X1, X7, X8, X9, X10 are weak, and each has four algebraic extraversions. Later, we shall meet points with up to ten extraversions. A line is strong if it has two strong points (or, equivalently, a strong tripole). A conic is strong if it has a strong perspector (or, equivalently, a strong centre). A pK cubic is strong if it has a strong pole and a strong pivot. We observe that, if X lies on a strong curve F, then all members of {X} are on F. We could equally define {X} as the intersection of all strong curves containing X. *** Definitions : A point P is an {X}cevian point if P is a centre in {X} with respect to its Cevian triangle. A point P is an {X}anticevian point if P is a centre in {X} with respect to its Anticevian triangle. See Table 45 for pedal equivalents. The most obvious example is that X2 is the only X2Cevian, and the only X2Anticevian point. We also know that
Examples (a), (b) and (c), (d) are related by the trivial Proposition 1 : Y is an {X}Cevian point if and only if X is a {Y}Anticevian point. Notations : If X is any point, X' denotes the complement of the isotomic conjugate of X. In other words, X' is the CTtransform of X or simply X' = ctX. The inverse is the TAtransform X" = taX, the isotomic conjugate of the anticomplement of X. Examples (a), (d) and (b), (c) are special cases of the more interesting Proposition 2 : Y is an {X'}Cevian point if and only if Y' is an {X}Anticevian point.
If a centre X is on a strong curve F, then we can define two loci: cF = { Y : Y is an {X}Cevian point }, and aF = { Y : Y is an {X}Anticevian point }. The condition that F is strong ensures that it contains all members of {X}. It also means that F can be defined using squares of sidelengths. Since the squares of the sidelengths of Cevian and Anticevian triangles can be computed easily, we can identify the curves cF and aF. For example, suppose that the cevian triangle of Y = x:y:z has sidelengths with squares P, Q, R, and that the coefficients of F are given in terms of a^2, b^2, c^2. Then we simply replace a^2, b^2, c^2 by P, Q, R and x, y, z by y+z, z+x, x+y in F(x,y,z) = 0 to get an equation for cF. Obviously aF is obtained similarly. *** A remarkable pencil of quintics Let m and n be two real numbers. Q[m,n] denotes the quintic m Q034 + n Q042. This pencil includes the Stothers Quintic Q012 = Q[1,2] and other quintics as seen below. Also, Q[1,1] is the union of the Kiepert Hyperbola and the sidelines. We have the result that the {X}Cevian points for X on a Q[m,n] lie on another member of the pencil. Likewise for the {X}Anticevian points. Proposition 3 : If F = Q[m,n], then cF = Q[3m+2n,2(m+n)], and aF = Q[2(m+n),2m+3n]. If F = Q012 = Q[1,2] or Q059 = Q[2,1], then F = cF = aF. Note that Q012 = Q[1,2] and Q059 = Q[2,1] contain all the associated Cevian and Anticevian points. This property is shared by the EulerMorley Quintic Q003. These and other interesting F, cF, aF are given in the following table. 



Note 1 : It follows that the Lucas Cubic contains
Note 2 : From Note 1 and Proposition 2, we see that the Thomson Cubic contains
Note 3 : This shows that the Kiepert Hyperbola contains the four {X3}Anticevian points. Note 4 : This shows that the EulerMorley Quintic Q003 contains
Note 5 : These two rows show that
Note 6 : The quintic Q[m,n] introduced above contains
Note 7 : The quintic Q[m,n] is circular if and only if m+2n = 0. This gives Q[2,1] = Q059.
*** Properties and special cases Each member of the pencil contains A, B and C, X2, X13, X14, Ma, Mb, Mc, the vertices of the Medial Triangle. The medians meet the curve again at the vertices of a triangle homothetic with ABC whose Avertex is m : m+n : m+n. This homothety has center G, ratio (2m+n)/(m+2n). Moreover
A distinct pair of members have 25 intersections, namely A, B, C, which count as 6 each, X2, which counts as two, X13, X14, Ma, Mb, Mc. Some special features Q012 = Q[1,2] has infinite points those of the Steiner Ellipse and the Thomson Cubic, the asymptotes at the latter concur at X6 meets X2X6 at X2 (twice), X6 and on the Steiner Ellipse, the tangents at Ma, Mb, Mc concur at X6. Q059 = Q[2,1] has infinite points those of the Circumcircle and the Medians, meets X2X6 at X2 (thrice) and on the Circumcircle. More generally, Q[m,n] contains the infinite points of the circumconic with perspector Z if and only if Z lies on the quartic Q062 which contains X2 (Steiner ellipse), X6 (Circumcircle), X523 (Kiepert hyperbola). Q095 = Q[1,1] is another interesting quintic of the pencil passing through the Steiner point X(99). 
