Tucker cubics and Bicentric Cubics

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1 Tucker cubics and icentric ubics ernard ibert reated : March 26, 2002 Last update : September 30, 2015 bstract We explore the configurations deduced from the barycentric coordinates of a given point and define the notion of bicentric triples, bicentric conics, bicentric cubics. 1 Playing with barycentric coordinates Let P = (p : q : r) be a point distinct of the centroid not lying on the sidelines of the reference triangle (i.e. such that pqr 0) and let t P = (1/p : 1/q : 1/r) be its isotomic conjugate. 1.1 Related points on the sidelines of The cevian triangle of P is denoted by P a P b P c with : P a = (0 : q : r), P b = (p : 0 : r), P c = (p : q : 0) The cevian triangle of t P is denoted by t P t ap t b P c with : t P a = (0 : r : q), t P b = (r : 0 : p), t P c = (q : p : 0) The parallels to at P b and to at P c meet at P ba and P ca respectively and, similarly, the parallels to at t P b and to at t P c meet at t P ba and t P ca respectively. The coordinates of these four points are : P ba = (0 : p : r), P ca = (0 : q : p), t P ba = (0 : r : p), t P ca = (0 : p : q) The corresponding points on the other sidelines of are defined likewise : P cb = (p : 0 : q), P ab = (q : 0 : r), P ac = (r : q : 0), P bc = (p : r : 0), t P cb = (q : 0 : p), t P ac = (q : r : 0), t P ab = (r : 0 : q) t P bc = (r : p : 0) We obtain a total of eighteen points on the sidelines corresponding to all the possible permutations of two non-zero coordinates taken among p, q, r. 1.2 Permuting coordinates We can now construct all the points obtained from P by permuting its coordinates. When one coordinate only stays at its place, we get the three following points : P p = (p : r : q), P q = (r : q : p), P r = (q : p : r) where P p = P cb P bc,p q = P ac P ca,p r = P ba P ab. Note that P p is also the intersection of the line t P with the parallel at P to the sideline. Their isotomic conjugates are obtained in the same way : t P p = (1/p : 1/r : 1/q) = (qr : pq : pr), 1

2 with t P p = t P cb t P bc for example. t P q = (1/r : 1/q : 1/p) = (pq : pr : qr), t P r = (1/q : 1/p : 1/r) = (pr : qr : pq) When none of the three coordinates stays in place, we obtain the two points : Q 1 = (q : r : p), Q 2 = (r : p : q) with Q 1 = t P r t P p t P q and Q 2 = t P q t P r t P p. These points Q 1, Q 2 are called the bicentric mates of P and P, Q 1, Q 2 is called a bicentric triple. Note that the centroid of a bicentric triple is always the centroid of. The line Q 1 Q 2 is parallel to the trilinear polar of t P and the midpoint of Q 1 Q 2 is the complement of P. Their isotomic conjugates are t Q 1 = (1/q : 1/r : 1/p) = (pr : pq : qr), t Q 2 = (1/r : 1/p : 1/q) = (pq : qr : pr). These two points are the perspectors of triangles and P ba P cb P ac, and P ca P ab P bc respectively. They are sometimes called the rocardians of P. Remark : the five triangles, P p P q P r, t P p t P q t P r, PQ 1 Q 2, t P t Q t 1 Q 2 have the same centroid and the three vertices of each triangle form a bicentric triple. Similarly, the eighteen points above also form six bicentric triples and the six corresponding triangles have the same centroid. We obtain some other points by intersecting other related lines : and similarly their isotomic conjugates R p = P r P q = (qr : q 2 : r 2 ), R q = P p P r = (p 2 : pr : r 2 ), R r = P q P p = (p 2 : q 2 : pq), t R p = t P r t P q = (qr : r 2 : q 2 ), t R q = t P p t P r = (r 2 : pr : p 2 ), t R r = t P q t P p = (q 2 : p 2 : pq). Note that thetriangles andr p R q R r are perspectiveat thepoint P 2 = (p 2 : q 2 : r 2 ), the barycentric square of P. 1.3 Perspectivities This configuration leads to a set of triply perspective triangles where the three perspectors are known points of the figure. This is summarized in the following table. triply perspective triangles and their three perspectors and P p P q P r t P, t Q 1 and t Q 2 and t P t p P t q P r P,Q 1 and Q 2 and PQ 1 Q 2 t P p, t P q and t P r and t P t Q t 1 Q 2 P p,p q and P r P p P q P r and R p R q R r t Q 1, t Q 2 and (p 2 +qr : q 2 +rp : r 2 +pq) t PpP t qp t r and t RpR t qr t r Q 1,Q 2 and (p+qr/p : q +rp/q : r+pq/r) 2

3 1.4 Special cases P = K (Lemoine point) This is the rocard configuration : t Q 1 and t Q 2 are the rocard points usually denoted by Ω 1 and Ω 2 and P p P q P r is the first rocard triangle P = H (orthocenter) This is the configuration cited in [5] where t Q 1 and t Q 2 are called cosine orthocenters (denoted by σ 1 and σ 2 in the Tucker paper) P = I (incenter) In this case t Q 1 and t Q 2 are sometimes called the Jerabek points. 2 Related conics 2.1 Steiner onics The six points P,Q 1,Q 2,P p,p q,p r lie on an ellipse E 1 centered at homothetic to the Steiner circumellipse with equation : (qr +rp+pq)(x 2 +y 2 +z 2 ) = (p 2 +q 2 +r 2 )(yz +zx+xy) Their isotomic conjugates t P, t Q 1, t Q 2, t P p, t P q, t P r lie on another ellipse E 2 centered at homothetic to the Steiner circum-ellipse with equation : pqr(p+q +r)(x 2 +y 2 +z 2 ) = (q 2 r 2 +r 2 p 2 +p 2 q 2 )(yz +zx+xy) Note that these two ellipses are bicentric conics since any point M = (x : y : z) on one ellipse has its two bicentric mates M 1 = (z : x : y) and M 2 = (y : z : x) on this same ellipse. See figure rocard onics The six points P, t Q 1, t Q 2,P p,p q,p r lie on a conic 1 centered at Ω 3 = [p(2qr +p(q +r p)) ::]. This conic also contains /P, the center of the circumconic with perspector P. Its equation is : qrx 2 +rpy 2 +pqz 2 = p 2 yz +q 2 zx+r 2 xy. 1 is a circle if and only if P = K (Lemoine point) and this circle is the rocard circle with diameter OK and center X 182. Similarly, the isotomic conjugates t P,Q 1,Q 2, t P p, t P q, t P r of the six points above also lie on a conic 2 centered at Ω 4 = [q 2 r 2 (qr p(q +r +2p)) ::]. Its equation is : pqr(px 2 +qy 2 +rz 2 ) = q 2 r 2 yz +r 2 p 2 zx+p 2 q 2 xy. This conic is the rocard circle if and only if P = X 76. These two conics are also bicentric conics. See figure 2. These two conics 1 and 2 generate a pencil that contains a circle if and only if P lies either on : 1. the line at infinity in which case 1 decomposes into the line at infinity and a line, 2. the Steiner ellipse in which case 2 decomposes into the line at infinity and a line, 3. the pivotal cubic K659 = pk(,k). In this latter case, the axes of the two conics have the same directions. 3

4 E 2 t Q1 t Pr P p Q 2 t Pq Q 1 P P q t P P r t Q2 t Pp E 1 Figure 1: The Steiner onics 2 t Q1 t Pr P p P Q 2 t Pq Q 1 P q t Q2 t P P r t Pp 1 Figure 2: The rocard onics 2.3 Three other conics P a,p b,p c, t P a, t P b, t P c lie on the bicevian conic 1 = (P, t P) with equation : pqr(x 2 +y 2 +z 2 ) = p(q 2 +r 2 )yz 4

5 1 is a circle if and only if the cyclocevian conjugate of P coincide with the isotomic conjugate t P of P. There are six such points which are the cevian O-points of Table 39 in [1]. P ab,p ba,p bc,p cb,p ca,p ac lie on the bicevian conic 2 = ( t Q 1, t Q 2 ) with equation : qrx 2 = (p 2 +qr)yz which is a circle if and only if P = X 194. Similarly t P ab, t P ba, t P bc, t P cb, t P ca, t P ac lie on the bicevian conic 3 = (Q 1,Q 2 ) with equation : pqr(px 2 +qy 2 +rz 2 ) = qr(p 2 +qr)yz which is a circle if and only if P is the isotomic conjugate X 2998 of X 194. Note that these three conics are bicentric conics again. See figure 3. P ac P c P cb P bc Q 2 t Q1 P P b t Pb t Pc t Pa t P P ba 1 2 Q1 P a t Q2 P ca P ab Figure 3: The onics 1 and 2 3 Tucker cubics 3.1 locus Let λ be a real number and let us denote by S the area of triangle. n easy computation shows that the locus of point M = (x : y : z) such that the area of the cevian triangle of M is constant and equal to λs is an isotomic non-pivotal circum-cubic T(λ) with equation : ( x(y 2 +z 2 ) ) xyz = 0 λ 5

6 or equivalently : or again : λ x(y 2 +z 2 )+2(λ 1)xyz = 0, ( 1 (x+y +z) x + 1 y + 1 ) = 1+ 2 z λ. Its root is the centroid. The equations above show that all these cubics are in a pencil of circum-cubics and also that each cubic is a bicentric cubic since any point M = (x : y : z) on the cubic has obviously its two bicentric mates M 1 = (z : x : y) and M 2 = (y : z : x) on the cubic. onsider the three conics γ, γ, γ with equations : γ : λ(x+y)(x+z) x 2 = 0, γ : λ(y +z)(y +x) y 2 = 0, γ : λ(z +x)(z +y) z 2 = 0. γ passes through, and is centered on the median at the image of in the homothety with center, ratio 2(λ 1). γ is tangent at, to the sidelines of the antimedial triangle. λ 4 γ, γ, γ are parabolas if and only if λ = 4. T (λ) is the jacobian of these three conics i.e. the locus of M such that the polar lines of M in the three conics concur at N which also lies on the cubic. The same locus is easily defined from a given point P = (p : q : r) as the locus of point M = (x : y : z) such that the (algebraic) areas of the cevian triangles of M and P are equal. The equation given above is rewritten under the form : pqr x(y 2 +z 2 ) = xyz p(q 2 +r 2 ) pqr x 2 (y +z) = xyz p 2 (q +r), the cubic being then denoted by T (P) and called the P-Tucker cubic since it generalizes a cubic found by R. Tucker in [5]. In this case, the Tucker cubic T(P) is clearly the non-pivotal isotomic cubic nk(,,p). Remark 1 : since two isotomic conjugates P and t P have equal area cevian triangles, we have T (P) = T( t P). Remark 2 : the third point of T (P) on the line through P and t P is Q and the tangents to the cubic at P and t P pass through t Q. The coordinates are : Q = p2 qr p(q r) : : ; t p(q r) Q = p 2 : :. qr Remark 3 : the locus of point M = (x : y : z) such that the (algebraic) areas of the cevian triangles of M and P are opposite is also a cubic T (P) with equation : pqr x(y 2 +z 2 ) = xyz p(q 2 +r 2 ) 2 (q +r). This cubic is also a non-pivotal isotomic cubic with pole and root. 6

7 3.2 Immediate properties Points on the curve It is obvious that T(P) passes through the eighteen following points : the vertices,, already mentionned. the points at infinity of the sidelines of which are inflexion points since the trilinear polar of the root is the line at infinity. P and its isotomic conjugate t P. the ten points obtained from P and t P by permuting their coordinates namely Q 1,Q 2,P p,p q,p r and their isotomic conjugates t Q 1, t Q 2, t P p, t P q, t P r. Recall that all these points come in triples formed by one point and its two bicentric mates symptotes They are obviously inflexional and parallel to the sidelines of. They form a triangle homothetic to at. The ratio of homothety is 2(λ 1). λ Tangents The tangent at,, are parallel to the sidelines of and are the sidelines of the antimedial triangle. From this, T (P) is tritangent to the Steiner circum-ellipse at these points. Each median meets T(P) again at two points and the tangents at them are also parallel to the sidelines of, this coming from the fact that the polar conic of the point at infinity of a sideline of degenerates into an asymptote and the relative median Polar conic of the centroid The polar conic of the centroid is generally an ellipse homothetic of the Steiner ellipse under the 4λ 1 homothety with center, ratio. Its equation is : λ 1 λ x 2 +(3λ 1) yz = 0 or λ (x 2 2yz)+(5λ 1) yz = 0 which clearly shows that this polar conic belongs to a pencil of concentric ellipses generated by the Steiner circum-ellipse and in-ellipse. It degenerates when λ = 1 (into the line at infinity counted twice) or when λ = 1/4 (into two imaginary lines through and the infinite points of the Steiner ellipse). When λ = 0 it is the Steiner circum-ellipse and when λ = 1/5 it is the Steiner in-ellipse. With λ = 1/3, we find the imaginary diagonal conic with equation x 2 +y 2 +z 2 = 0. When λ [0;1/4[ ]1;+ [, the polar conic is a real non-degenerate ellipse. Furthermore, if λ ]1;+ [ the polar conic of meets the cubic at six real points hence it is possible to draw six real tangents from to the cubic. See for example the Tucker-Poncelet cubic below. 3.3 onstruction of T (P) The parallel to at P intersects at and the parallel to at P intersects at. For any point m on the line, let us denote by m its homologue under the involution (on ) which swaps, and,. n easy way to realize the construction of m is the following : the line P intersects again the circle at E and the circle Em meets again at m. 7

8 The conic through,,, t P, t m (which is the isotomic conjugate of the line Pm) intersects the line Pm at two points on the cubic T (P). 3.4 Prehessians of T (P) EverycubicT (P)hasthreeprehessiansandoneofthemisalways real witharemarkablysimpleequation, namely : (q +r)(r +p)(p+q) (x 3 +y 3 +z 3 ) = 3pqr (y +z)(z +x)(x+y). It is interesting to observe that the mapping that sends any point of the plane to the center of its polar conic in this prehessian is the isotomic conjugation. 4 Some special examples of T (P) Recall that T(P) is actually the non-pivotal isotomic cubic nk(,,p). 4.1 The original Tucker cubic K011 This cubic is T(H) (or T( t H) where t H = X 69 ) and is the one quoted in [5]. It contains X 4, X 69, X 877, X The Tucker-rocard cubic K012 This is T (K) (or T ( t K) where t K = X 76 ) passing through the rocard points and the vertices of the first rocard triangle. It contains X 6, X 76, X 880, X The Tucker-ergonne-Nagel cubic K013 This is T (X 7 ) or T (X 8 ) where X 7, X 8 are the ergonne and Nagel points respectively. It contains X 7, X 8, X 883, X The Tucker-Jerabek cubic K014 This is T(I) (or T( t I) where t I = X 75 ) passing through the Jerabek points. It contains X 1, X 75, X 874, X T (P) with concuring asymptotes K016 There is one and only one non-degenerate T (P) with concuring asymptotes (at ) obtained when λ = 1 : this is the locus of point M such that the area of the cevian triangle of M is equal to S. 4.6 The Tucker cubic K015 T (P) is unicursal ifand only if it passesthrough oneandonly onefixedpoint of the isotomic conjugation. When this fixed point is one of the vertices of the antimedial triangle, the cubic decomposes into the union of the sidelines of this triangle. Hence there is one and only one unicursal Tucker cubic and it is T() with a singularity at : T () is a conico-pivotal isocubic with pivotal-conic the Steiner circum-ellipse i.e. for any point M on the curve, the line through M and its isotomic conjugate t M envelopes the Steiner circum-ellipse. The equation of T () is : x(y 2 +z 2 ) 6 xyz = 0 8

9 4.7 The Tucker-Poncelet cubic K327 This is T(2), locus of point M such that the area of the cevian triangle of M is 2S. It is an example of Tucker cubic such that six real tangents can be drawn through to the cubic. See figure 4. polar conic of Figure 4: The Tucker-Poncelet cubic K327 5 family of related cubics 5.1 The cubics V(P) Recall that the equation of the Tucker cubic T (P) can be written under the form pqr x 2 (y +z) xyz p 2 (q +r) = 0. permutation of the plus and minus signs in this equation gives pqr x 2 (y z)+xyz p 2 (q r) = 0, (1) or equivalently p(y z)+ (q r)x = 0, (2) and this leads us to a new bicentric cubic V(P) that contains many points related with T(P). This cubic is a bicentric cubic again and it is in fact the pseudo-pivotal cubic psk(,, t P) in [2]. Indeed, V(P) and T (P) have nine identified common points namely,,, t P, P p, P q, P r, t Q 1 and t Q 2. Note that the tangents at,, are the medians of. Recall that T (P) is a self-isotomic non-pivotal cubic. On the other hand, the isotomic transform of V(P) is V( t P) with equation p(y z) (q r)x = 0. (3) 9

10 which is clearly the pseudo-pivotal cubic psk(,, P). See [2] for further properties. In particular, V(P) is globally unchanged under two inverse transformations which are the product in either ways of isotomic conjugation and eva conjugation. Moreover, V(P) also contains : the midpoints M a, M b, M c of. /P (the eva conjugate of P, the center of the circumconic (P) with perspector P) and its two bicentric mates. c P (the cross conjugate of P) and its two bicentric mates. t Q 0, the tangential of t P, the intersection of the lines t Q 1, t Q 2 and /P, c P. See figure 5. t Q0 P Mc t Q1 Pq P Ma Pr t P t Pp Q2 Mb /P V(P) T(P) Figure 5: V(P) and T(P) the infinite points of the pivotal cubic K 1 with pole P and pivot the anticomplement of t P i.e. the P eva conjugate of. This easily derives from the following form of equation (1) : ( qr +rp+pq)x(ry 2 qz 2 ) = (x+y +z) p 2 (q r)yz. V(P) and K 1 meet at six other points on the circumconic 1 (P) with equation p 2 (q r)yz = 0. These points are,,, c P and two points P 1, P 2 on the line, t P. Note that 1 (P) also contains P, the complement of t P and P 2, the barycentric square of P, K 1 is a cubic with three concurring asymptotes. See figure 6. One remarkable thing to observe is 10

11 1(P) t P P1 Mc P Ma P2 Mb V(P) K1 Figure 6: V(P) and K 1 that the asymptotes of V(P) are inflexional. Indeed, the hessian cubic of V(P) has equation (q r) p(y z)+ (q r)x = 4p 2 q 2 r 2 (x+y +z)(x 2 xy +y 2 xz yz +z 2 ), wherethelefthandsiderepresentstheequation of V(P) itself andwhere(x 2 xy+y 2 xz yz+z 2 ) = 0 is the union of the two imaginary lines passing through and the infinite points of the Steiner ellipse. It follows that the poloconic of the line at infinity must be an ellipse with center, homothetic to the Steiner ellipse and inscribed in the triangle formed by the asymptotes. the common points,,, M 1, M 2, M 3 of the Steiner ellipse and the pivotal cubic K 2 with pole P and pivot /P. The tangents at these six points concur at since the polar conic of in V(P) is the Steiner ellipse. Note that this latter cubic has three asymptotes concurring at. Furthermore, each asymptote of V(P) is parallel to a line passing through one vertex of and one vertex of M 1 M 2 M 3. See figure 7. V(P) and K 2 meet at three other points on the line P,/P namely /P, P 3, P 4. See figure 8. This is obtained with another form of (1) : p( p+q +r)x(ry 2 qz 2 ) = (yz +zx+xy) qr(q r)x. the common points of (P) and the pivotal cubic K 3 with pole P and pivot t P. The three remaining common points are t P and the two points P 1, P 2 on the line, t P. See figure 9. This is obtained with another form of (1) : qrx(ry 2 qz 2 ) = (pyz +qzx+rxy) p(q r)x. 11

12 V(P) M2 M3 Mc Mb Ma M1 Figure 7: V(P) and its asymptotes t P M3 P1 Mc P4 P Ma /P P2 Mb M2 K2 M1 P3 V(P) Figure 8: V(P) and K 2 the common points M a, M b, M c, T a, T b, T c of the inscribed Steiner ellipse and the complement K 4 of the pivotal cubic K 2 with pole P and pivot /P. The three remaining common points are /P and the two intersections P 3, P 4 of the line P,/P with the circumconic passing through and t P, namely the isotomic transform of the line,p. 12

13 K3 t P M3 P1 Mc P P4 Mb M2 Ma /P P2 M1 P3 V(P) (P) Figure 9: V(P) and K 3 Note that the three points T a, T b, T c are the midpoints of M 1 M 2 M 3. This is also obtained with another form of (1) : (2yz x 2 ) qr(q r)x = 4(y z)(z x)(x y) qr(q r)x( x+y +z) 2, where 4(y z)(z x)(x y) qr(q r)x( x+y +z) 2 = 0 is the equation of K 4. See figure 10. onstruction of V(P) : Let H P be the diagonal conic through P, /P, the vertices of the anticevian triangles of P and /P, L P be a variable line through P meeting H P again at Q, P be the isotomic transform of L P. The line Q,/P meets P at two points of V(P). Example : the rocard-van Tienhoven cubic V(K) = K512 This cubic was brought to my attention by hris van Tienhoven in a private message. It is probably the most interesting cubic V(P) since it is closely related to the rocard geometry and naturally to the Tucker-rocard cubic K012. Indeed, these two cubics contain the rocard points Ω 1, Ω 2, the vertices of the first rocard triangle, the third rocard point X 76, O = /K and X 3224 = c K. See figure 11. Furthermore, V(K) meets 13

14 K4 t P M3 T1 P1 T2 Mc P4 P /P Ma T3 P2 Mb M2 M1 P3 V(P) Figure 10: V(P) and K 4 K012 Ω1 K X76 O Ω2 Steiner ellipse K512 Figure 11: The rocard-van Tienhoven cubic and K012 the line at infinity at the same points as pk(x 6,X 194 ) = K410, the Steiner ellipse at the same points as the Mcay cubic K003, the circumcircle of at the same points as pk(x 6,X 76 ) = K184. See figure

15 K184 Ω1 K X76 O Ω2 K512 K003 K410 Figure 12: The rocard-van Tienhoven cubic and K003, K184, K Pencils generated by V(P) and T(P) Recall that V(P) and T(P) have nine identified common points hence they generate a pencil F(P) of cubics passing through these nine points. F(P) contains several other remarkable cubics. one cubic V + (P) without term in xyz (a K 0 cubic) which is at the same time a K + i.e. having three concurring asymptotes, here in. See figure 13. The asymptotes of V + (P) are also inflexional. Indeed, the hessian cubic of V + (P) has equation which we have already met above. (x+y +z)(x 2 xy +y 2 xz yz +z 2 ) = 0 one cubic V (P) inscribed in the antimedial triangle a b c. This also contains the anticomplement of the isotomic conjugate of the complement of P. The tangents at a, b, c pass through. See figure eneral bicentric circum-cubics It is obvious that a bicentric circum-cubic u,v,w must have an equation of the form : u(x 2 y +y 2 z +z 2 x)+v(xy 2 +yz 2 +zx 2 )+4wxyz = 0, where u, v, w are any three real (not all zero) numbers. Note that u,v,w meets the sidelines of at three points independent of w namely : U = (0 : v : u), V = (u : 0 : v) and W = ( v : u : 0). 15

16 Pr t P Mc Mb t Q2 Pp Ma t Q1 Pq T(P) V+(P) V(P) Figure 13: V + (P), V(P) and T (P) c Pr t P b Mc Mb t Q2 Pp Ma t Q1 Pq a V (P) T(P) V(P) Figure 14: V (P), V(P) and T (P) These points form a bicentric triple with centroid and, moreover, they are the tangentials of the vertices of i.e. the tangents at,, to u,v,w pass through U, V, W respectively. More generally, for any point P on u,v,w, the triangle formed by P and its bicentric mates is 16

17 inscribed in the cubic and rotates around when P traverses the cubic. The isotomic transform of u,v,w is clearly v,u,w and this shows that u,v,w is a self-isotomic cubic if and only if u = v. These are the Tucker cubics. This is the only case when the points U, V, W are collinear. The points U, V, W are the vertices of a cevian triangle if and only if u + v = 0, and with w suitably chosen, we obtain the cubics V(P) seen above. In the particular case w = 0, we obtain V + (P). 6.1 Polar conic of the centroid The polar conic u,v,w of with respect to u,v,w can be written under the form : (u+v)(x+y +z) 2 +4w(yz +zx+xy) = 0, (4) showing that it is generally an ellipse with center homothetic to the Steiner ellipse. This can be real or not, proper or degenerate. The ratio ρ of homothecy is given by ρ 2 3u+3v +4w =. 4w Naturally u,v,w meets u,v,w at six points with tangents concurring at. The equation (4) and the ratio ρ clearly confirm that : when u+v = 0, u,v,w is the Steiner ellipse (ρ = 1), when w = 0, u,v,w has three inflexional asymptotes concurring at since u,v,w is the line at infinity counted twice (ρ = ), when u+v +w = 0, u,v,w is the Steiner in-ellipse (ρ = 1/2), when u+v = 4w, u,v,w is the Steiner ellipse of the antimedial triangle (ρ = 2), when u+v +2w = 0, u,v,w is the diagonal conic x 2 +y 2 +z 2 = 0 (ρ 2 = 1/2). Each condition above leads to a pencil of cubics such that the polar conic of is the same for any cubic in the pencil. 6.2 icentric circum-cubics passing through a given point Let us now impose that the cubic u,v,w contains a given point P. It is clear from the general equation above that the corresponding cubics now form a pencil passing through nine fixed points forming three bicentric triples. These triples are,, P, Q 1, Q 2 t P p, t P q, t P r. Each cubic of the pencil can be characterized by a single parameter and, when this parameter varies, the polar conic of meets the corresponding cubic at six points lying on a same bicentric circum-quintic Q(P) that contains P, Q 1, Q 2, t P p, t P q, t P r, the infinite points of the Steiner ellipse,, a, b, c, the anticomplement ap of P and its two bicentric mates. This quintic has five asymptotes concurring at. Note that the tangents at,,, P, Q 1, Q 2, t P p, t P q, t P r pass through. See figure Examples We conclude this paper with several selected examples and some additional remarks concerning each particular situation.. 17

18 c b tpp P Q1 tpq Q2 tpr ap a Figure 15: The bicentric circum-quintic Q(P) Example 1 When u,v,w is the Steiner ellipse, the cubic u,v,w must pass through the midpoints of and thus the tangents at,, must be the medians of. u,v,w meets the Steiner ellipse at three other points,, forming a bicentric triple with centroid. ll these cubics form a pencil and the cubic that passes through the given point P = (p : q : r) has equation : px(ry qy)(qy+rz) = 0. It also contains the eva conjugate of t P and the crossconjugate of t P (equivalently the isotomic conjugate of the eva conjugate of P) and the four corresponding bicentric mates. The tangential of P is the third point on the line passing through the two former points. The figure 16 shows the cubic that contains X 9, X 75, X 87 and naturally their six bicentric mates Example 2 When u,v,w is the Steiner in-ellipse, the cubic u,v,w passing through P meets this ellipse at six (real or not) points with their tangents passing through. These six points form two other bicentric triples. The figure 17 shows the cubic that contains X 76, the rocard points, the vertices of the first rocard triangle. 18

19 ' ' X87 X75 X9 ' Figure 16: u,v,w with polar conic the Steiner ellipse X76 Ω1 1 K 1 Ω2 1 O Figure 17: u,v,w with polar conic the Steiner in-ellipse Example 3 When u,v,w is the union of the two imaginary lines passing through and the infinite points of the Steiner ellipse, the cubic u,v,w is an acnodal cubic with as isolated node on the curve. 19

20 The figure 18 shows the cubic that contains X 76, the rocard points, the vertices of the first rocard triangle. 1 Ω1 K 1 X76 1 O Ω2 Figure 18: u,v,w with a degenerated polar conic Example 4 When u,v,w is the Steiner ellipse of the antimedial triangle a b c, the cubic u,v,w passing through P also contains a, b, c and naturally Q 1, Q 2, t P p, t P q, t P r, these nine points forming three bicentric triples. Figure 19 shows the cubic that contains X

21 c b tpp Q1 X69 tpr tpq Q2 a Figure 19: u,v,w with polar conic the Steiner ellipse of the antimedial triangle 7 eneralized Tucker cubics The Tucker cubics T(P) = nk(,,p) we met above are deeply related to the centroid of triangle through isotomic conjugation and therefore to the line at infinity, the trilinear polar of that contains the three real inflexion points of T (P). The corresponding cubics V(P) have the same relationship with since they are pseudo-pivotal cubics with pseudo-pole and pseudo-pivot. See [2] for more details. Recall that both cubics are bicentric cubics. In this section, is replaced by any other point Z not lying on one sideline of and the line at infinity by the trilinear polar L Z of Z meeting the sidelines of at U, V, W. The isotomic conjugation is now the isoconjugation with fixed points Z and its three harmonic associates. The isoconjugate of any point M is still denoted by t M. ll the points defined at the beginning are here defined similarly since the parallels to the sidelines of are replaced by lines intersecting on L Z. More precisely, the parallels at M to,, are now the lines UM, VM, WM respectively. The Z Tucker cubic T (Z,P) = nk(z 2,Z,P) is now : ( x u + y v + z ) ( u w x + v y + w ) ( p = z u + q v + r ) ( u w p + v q + w ) r pqr vwx 2 (wy +vz) xyz p 2 vw(wq+vr) = 0, and the corresponding cubic V(Z,P) = psk(z 2,Z, t P) is : pqr vwx 2 (wy vz)+xyz p 2 vw(wq vr) = 0 pqr (wy vz)+xyz (wq vr) = 0, 21

22 where Z 2 denotes the barycentric square of Z. Note that these two cubics are not bicentric cubics anymore. References [1] ibert. ubics in the Triangle Plane, available at [2] ibert. Pseudo-Pivotal ubics and Poristic Triangles, available at [3] Kimberling., Triangle enters and entral Triangles, ongressus Numerantium, 129 (1998) [4] Kimberling., Encyclopedia of Triangle enters, [5] Tucker R., The cosine orthocentres of a triangle and a cubic through them. [J] Mess. (2) XVII , (1887). 22