Triangle conics and cubics

This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article was to develop an algorithm for creating curves, which pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This was the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new intersting geometrical object. During the research were derived number of new triangle conics and cubics, were considered their properties in Euclidian space. In addition, was discussed corollaries of the obtained theorems in projective geometry, what proves that all of the descovered results could be transfered to the projeticve plane.


Base triangle
Excentral triangle I GOT (homothetic center k of the orthic and tangent triangles) Table 1. Correspondence table between points of the base and excentral triangles All of the above facts could be easily proved by basic principles of classiical geometry [1]. Hence, we may apply derived results for creating new triangle cubics and conics. Firstly Jerabek hyperbola was considered.
We may observe that Jerabek hyperbola for the extcentral tringle has number of points which corresponde to other ones in the base triangle. The study of such matches gave us significant results.
Jerabek hyperbola for excentral triangle New hyperbola for the base triangle   Similarly we studied Thomson cubic for the base trianlge and matched its points with ones in the excetral trianlge.
By apllying correspondence table between points of the base and excentral triangles to the points of Thomson cubic we obtain a new triangle cubic. GOT (gomotetic center of the orthic anf tngent triangles)   Definition 3. Darboux cubic is a curve that passes through vertices of the triangle, centers of the excircles, incenter, circumcenter, Bevan point [3].
Trianlge centers of the Darboux cubic in the base triangle were matched with points in the excentral triangle.
Darboux cubic for the base triangle New cubic for the excentral triangle A, B, C (vertices of the base triangle) H 1 , H 2 , H 3 (bases of the altitudes)  The discussed above results were obtained from considering excentral triangle, its triangle centers and correspondence between points in the excentral and basic triangle. As a result, were derived three new triangle curves, which were not discovered before. However, to get wider results were applied the same idea to other triangles. Namely was consider medial triangle. In the same way as before, was proven the fact that some points in the medial triangle match with some points in the base triangle. Proof of the mentioned facts relies on the patterns of the Euclidean geometry, some of the correspondence were proved before [1].
Points in the medial triangle Point in the base triangle Sy (Lemoine point)  We got new conic, which has vertices in the centroid and de Longchaps point, focus in the orthocenter. Directix of the Yff hyperbola is perpendicular to the Euler line and passes through center of the Euler circle. Euler line for the medial and base triangles coincide. However, center of the Euler circle of the base triangle is circumcenter for the medial. Therefore, directrix of the new hyperbola is perpendicular to the Euler line and passes through circumcenter.    We make the correspondence between points of the Lucas cubic in the medial triangle with triangle centers in the base triangle. As a result we obtain the following table.
Therefore, while applying correspondence method to the medial triangle we derived one new conic and two new cubics. In addition, we obsereved Euler and mid-arc triangles as correspondence base.

Lucas cubic for the medial triangle
New cubic for the base triangle         Based on the correspondence between point sof the mid-arc and base triangles we discovered new cubic which is based on Jerabek hyperbola.
Jerabek hyperbola for mid-arc triangle New hyperbola for the base triangle    Since, geometry of conic sections and other triangle curves are broadly used in the projective geometry we looked on the obtained result through the prism of the projective geometry.
According to the Pascal's theorem if six arbitrary points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet at three points which lie on a straight line.
Let's consider the first derived triangle curve based on Jerabek hyperbola for the excentral triangle. New hyperbola passes though excenters, Bevan point, incenter, mittenpunkt, de Longchaps point. We bulit a hexagon with verices in the given triangle centers and apply Pascal's theorem.
Let I 1 , I 2 be excenters, and Be, M i, L be Bevan point, M i mittenpunkt, de Longchamps point, respectively. We get the following results: Corollary 9. Concurent points of I 2 Be and LI, BeM i and I 1 L, M iI and I 2 I 1 belong to one line.
Corollary 10. Concurent points of segments I 2 M i and BeI 1 , BeL and II 2 , M iL and II 2 lie on one line.
Simiraly, we have applied the same idea for the hexagon inscribed in the new hyperbola derived from the Jerabek hyperbola for the mid-arc triangle.
Let A 2 , A 3 be middles of the arcs of the circumcircle, and Be, I, S, O be Bevan point, incenter, Speaker point, circumcenter, respectively. The following facts were discovered: Corollary 11. Points of intersection of lines A 2 Be and SI, IA 3 and OA 2 , BeA 3 and OS belong to one line.
Corollary 12. Points of intersection of line segments BeO and A 3 A 2 , BeS and IA 2 , A 3 S and IA 2 belong to a straight line.
Moreover, combination of two of the discovered triangle cubics gives us very interesting corollary as well. Let's consider new cubic derived from the Darboux cubic for the excentral triangle and new cubic constructed with the base Darboux cubic with respect to the medial triangle. The first mentioned new cubic passes through bases of altitudes, vertices, orthocenter, nine-point center, circumcenter, let's name it P (x, y). The second mentioned new cubic passes through middles of the triangle sides, Speaker point, nine-point center, circumcenter, orthocenter, let's name it Q(x, y). We may notice that this two cubics pass through three common points which are nine-point center, circumcenter and orthocenter. Moreover, this three points belong to Euler line, let it has an equation ax + by + c = 0. Since, we have two cubics which pass through points which belong to one line, there exists such integer t such that the following holds: P (x, y) − tQ(x, y) . . . ax + by + c. Therefore, Euler line is linear component of the composition of two new cubics. In addition, points of intersection of the linear component with the curve are inflection points [4].
Corollary 13. Euler line is a linear component of the composition of new triangle cubic (passes through bases of altitudes, vertices, orthocenter, nine-point center, circumcenter) and new triangle cubic (passes through middles of the triangle sides, Speaker point, nine-point center, circumcenter, orthocenter). Moreover, orthocenter, circumcenter, and nine-point center are inflection point of the composition of these two curves.
The above corollaries prove that the discovered in the research new triangle curves could be applied in different geometric areas and studied in advanced.
Remark 14. A further continue of our research consists in the same analysis of singularities as provided by second author in [6,8] for cubic obtained by us in the presented work.
Conclusion. During the research were discovered three new triangle conics and five new triangle cubics, what is very significant result for the classical geometry. In addition, was shown that proceedings of the study could be applied not only in Euclidian space, but in projective as well. However, the main result of the research was developed algorithm of deriving new triangle curves. This opens an opportunity for creating more triangle curves, while applying the method for various triangles, points, and geometric constructions.
The developed idea significantly simplifies the question of creating curves passing through triangular centers. However, it opens up a number of new questions. Which interesting properties do new curves have? What is the topological nature of these transformations? Is it possible to apply a similar idea to non-Euclidean objects? Could one use the same method over an arbitrary finite field? Can this idea be further generalized?