TTRIANGLE CONICS AND CUBICS
Veronika StrarodubAmerican University in Bulgaria, e-mail: [email protected]
Ruslan SkuratovskiiThe National Technical University of Ukraine Igor Sikorsky Kiev Polytechnic Institute, e-mail: [email protected], [email protected]
Abstract.
This is a paper about triangle cubics and conics in classical geometry with elementsof projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic usingan 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 differenttriangle centers in distinct triangles coincide. The simplest example: an incenter in a basetriangle 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 anothertriangle. Therefore, we get a new intersting geometrical object. During the research werederived number of new triangle conics and cubics, were considered their properties in Euclidianspace. 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.
Key words: classical geometry, triangle cubics, conics and cubics. MSC: 14H50Firstly, we will consider excentral triangle. Correspondence between points of the excentraland base triangles will give us significant results in developing new triangle curves. Below youmay observe correspondence table between points of the base and excentral triangles.
Base triangle Excentral triangle I (incenter) H (orthocenter) O (circumcenter) E (nine-point center) Be (Bevan point) O (circumcenter) Mi (mittenpunkt) Sy (Lemoine point) Mi (cid:48) (isogonal conjugate of themittenpunkt with respect to thebase triangle) M (centroid) Sp (Speaker point) Ta (Taylor point) Sy (Lemoine point) Sy( H H H ) (Lemoine point ofthe orthic triangle) Mi (cid:48)(cid:48) (isogonal conjugate point ofthe mittenpunkt with respect tothe excetntral triangle) GOT (homothetic center k of theorthic and tangent triangles)
Table 1.
Correspondence table between points of the base and excentral trianglesAll 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. FirstlyJerabek hyperbola was considered.
Definition 1.
Jerabek hyperbola is a curve which passes through verticies of trianlge, circum-center, orthocenter, Lemoine point, isogonal conjugate of the de Longchamps point [5]. a r X i v : . [ m a t h . M G ] S e p TRIANGLE CONICS AND CUBICS
We may observe that Jerabek hyperbola for the extcentral tringle has number of points whichcorresponde to other ones in the base triangle. The study of such matches gave us significantresults.
Jerabek hyperbola for excentral triangle New hyperbola for the base triangle
A, B, C (vertices of the base tringle) I , I , I (excenters) O (circumcenter) Be (Bevan point) H (orthocenter) I (incenter) Sy (Lemoine point) M i (mittenpunkt) L (cid:48) (isogonal conjugate of the de Longchamps point) L (de Longchamps point) Table 2.
Mtaching points for Jerabek hyperbolaTherefore, we got new triangle hyperbola 1, which passes through cneters of the excircles,Bevan point, incenter, mittenpunkt and de Longchamps point. It is still rectangular as Jerabekhyperbola is. Known fact about Jerabek hypperboal is that its center is center of the Eulercircle. However, Euler circle of the excentral trinangle is circumscribed circle for the base tri-angle. In addition, Jerabek Hyperbola is isogonaly conjugate to the Euler line. Meanwhile,Euler line for the excentral triangle is line ( I (incenter), O (circumcenter), Be (Bevan point), M i (cid:48) (isogonal conjugate for the mittenpunkt with respect to the extriangle),
M i (cid:48)(cid:48) (isogonal con-jugate for the mittenpunkt with respet to the base triangle)). Therefore, we can conclude, thatour new hyperbola is isogonaly conjugate to the line (
I, O, Be, M i (cid:48) , M i (cid:48)(cid:48) ) and its center is thecircumcenter.
Theorem 1.
New hyperbola passes though excenters, Bevan point, incenter, mittenpunkt, deLongchaps point. It is isogonal conjugate to the line ( I, O, Be, M i (cid:48) , M i (cid:48)(cid:48) ) , and its center iscircumcircle. Figure 1.
New trianlge conic based on Jerabek hyperbola for the excentral triangleSimilarly we studied Thomson cubic for the base trianlge and matched its points with onesin the excetral trianlge.
Definition 2.
Thomson cubic is a curve that passes through verticies of the trianlge, mid-dles of triangle sides, centers of the excircles, incenter, centroid, circumcenter, Lemoinepoint,mittenpunkt, isogonal conjugate of the mittenpunkt [1].
RIANGLE CONICS AND CUBICS 3
By apllying correspondence table between points of the base and excentral triangles to thepoints of Thomson cubic we obtain a new triangle cubic.
Thomson cubic for the base triangle New cubic for the excentral triangle
A, B, C (vertices of the base trianlge) H , H , H (bases of the altitudes) M a , M b , M c (middles of the base triangle sides) M ha , M hb , M hc (midles of orthic triangle’s sies) I , I , I (excenters) A, B, C (vertices) I (incenter) H (orthocenter) M (centroid) M ( H H H ) (centroid in the orthic triangle) O (circumcenter) E (nine-point center) Sy (Lemoine point) Sy ( H H H ) (Lemoine point of the orthic triangle) M i (mittenpunkt) Sy (Lemoine point) M i (cid:48)(cid:48) (isogonal conjugate of themittenpunkt with respet to theexcentral triangle)
GOT (gomotetic center of the or-thic anf tngent triangles)
Table 3.
Mtaching points for Thomson cubicAccording to the table, we got new cubic.
Theorem 2.
New triangle cubic 2 passes through vertices of the triangle, bases of the altitudes,middles of the triangle sides in the orthic triangle, orthocenter, Euler point, centroid in orthictriangle, Lemoine point, and gomotetic center of the orthic and tangent triangles.
Figure 2.
New trianlge conic based on Thomson hyperbola for the excentral triangleAnalogically was derived new trianlge cubic based on the Darboux cubic and correspondenceof its points with triangle centers in the excentral triangle.
Definition 3.
Darboux cubic is a curve that passes through vertices of the triangle, centers ofthe excircles, incenter, circumcenter, Bevan point [3].
TRIANGLE CONICS AND CUBICS
Trianlge centers of the Darboux cubic in the base triangle were matched with points in theexcentral triangle.
Darboux cubic for the base triangle New cubic for the excentral triangle
A, B, C (vertices of the base triangle) H , H , H (bases of the altitudes) I , I , I (excenters) A, B, C (vertices) I (incenter) H (orthocenter) O (circumcenter) E (nine-point center) Be (Bevan point) O (circumcenter) Table 4.
Mtaching points for Darboux cubic
Theorem 3.
Therefore, we got new cubic 3, which passes through vertices of the triangle, basesof the altitudes, orthocenter, Euler center, and circumcenter.
Figure 3.
New trianlge conic based on Darboux cubic for the base triangle incorrespondence with excentral trianlgeThe discussed above results were obtained from considering excentral triangle, its trianglecenters and correspondence between points in the excentral and basic triangle. As a result, werederived three new triangle curves, which were not discovered before. However, to get widerresults were applied the same idea to other triangles. Namely was consider medial triangle.
Definition 4.
Medial triangle is a triangle with vertices in the middles of the base trainglesides.In the same way as before, was proven the fact that some points in the medial triangle matchwith some points in the base triangle. Proof of the mentioned facts relies on the patterns ofthe Euclidean geometry, some of the correspondence were proved before [1].
Definition 5.
Yff hyperbola is a triangle curve wich passes through centroid, orthocenter,circumcenter, and Euler center.
RIANGLE CONICS AND CUBICS 5
Points in the medial triangle Point in the base triangle I (incenter) Sp (Speaker point) M (centroid) M (centroid) O (circumcenter) E (nine-point center) H (orthocenter) O (circumcenter) L (de Longchamps point) H (orthocenter) Be (Bevan point) Be ( M M M ) (Bevan point ofthe medial triangle) N a (Nagel point) I (incenter) G (Gergonne point) M i (mittenpunkt) Sy A (Lemoine point of the anti-complemetary triangle) Sy (Lemoine point) B (third Brocard point) M B (Brocard midpoint) Table 5.
Correspondence table between points of the medial and base triangles
Yff hyperbola for the base triangle New hyperbola for the medial traingle M (centroid) M (centroid) H (orthocenter) L (de Longchaps point) O (circumcenter) H (orthocenter) E (nine-point center) O (circumcenter) Table 6.
Mtaching points for the Yff hyperbola and medial triangleWe have considered Yff hyperbola for the base triangle and matched its point with trianglecenters of the medial triangle, applying correspondence table.We got new conic, which has vertices in the centroid and de Longchaps point, focus in theorthocenter. Directix of the Yff hyperbola is perpendicular to the Euler line and passes throughcenter of the Euler circle. Euler line for the medial and base triangles coincide. However, centerof the Euler circle of the base triangle is circumcenter for the medial. Therefore, directrix ofthe new hyperbola is perpendicular to the Euler line and passes through circumcenter.
Theorem 4.
New conic 4 is a curve that has vertices in the centroid and de Longchaps point,focus in the orthocenter. Directrix of the new hyperbola is perpendicular to the Euler line andpasses through circumcenter.
Figure 4.
New conic based on Yff hyperbolaLet‘s go further in our research and create more triangle cubics with the help of correspon-dence between triangle centers in medial and base triangles.
TRIANGLE CONICS AND CUBICS
Darboux cubic for the medial triangle New cubic for the base triangle
A, B, C (triangle vertices) M , M , M (middles of the triangle sides) A , B , C (antipods of the triangle) Antipods of the medial triangle I (incenter) Sp (Speaker point) O (circumcenter) E (nine-point center) H (orthocenter) O (circumcenter) L (de Longchamps point) H (orthocenter) L (cid:48) (isogonal conjugate of the de Longchaps point) H A (complementary conjugate of the orthocnter) Table 7.
Mtaching points for the Darboux cubic and medial triangleWe obsereved the transformation of the Darboux cubic and under the correspondence. It ledus to a new cubic.
Theorem 5.
We got new cubic 5, which passes through Speaker point, center of the Eulercircle, circumcenter, orthocenter, complementary conjugate of the orthocenter, middles of thetriangle side, and antipodes of the medial triangle.
Figure 5.
New cubic based on Darboux cubic for the medial triangleSimiraly, we take Lucas cubic for the base cubic and match it with points of the medialtriangle.
Definition 6.
Lucas cubic is such curve which passes through triangle vertices, orthocenter,Gergone point, centroid, Nagel point, Lemoine point of the anticomplementary tringle, andvertices of th anticomplementary triangle.We make the correspondence between points of the Lucas cubic in the medial triangle withtriangle centers in the base triangle. As a result we obtain the following table.
Theorem 6.
New cubic 6 which passes through Lemoine point, centroid, circumcenter, mit-tenpunkt, incenter, orthocenter, triangle vertices, and middles of the triangle sides.
Therefore, while applying correspondence method to the medial triangle we derived onenew conic and two new cubics. In addition, we obsereved Euler and mid-arc triangles ascorrespondence base.
RIANGLE CONICS AND CUBICS 7
Lucas cubic for the medial triangle New cubic for the base triangle Sy A (Lemoine point of the anticomplementary triangle) Si (Lemoine point) M (centroid) M (centroid) H (orthocenter) O (circumcenter) Ge (Gergonne point) M i (mittenpunkt)
N a (Nagel point) I (incenter) L (de Longchamps point) H (orthocenter) Table 8.
Mtaching points for the Lucas cubic and medial triangle
Figure 6.
New cubic based on Lucas cubic for the medial triangle
Definition 7.
Euler triangle is triangle with vertices in the intersection points of the trianglealtitudes and nine-point circle.
Definition 8.
Mid-arc triangle is a triangle with vertices in the middles of the arcs of thecircumcircle.Let’s firstly consider correspondence of points between Euler and base triangles.
Points in the Euler triangle Points in the base triangle I (incenter) M IH (midpoint of incener and orthocenter) M (centroid) M MH (midpoint of centroid and orthocenter) O (circumcenter) E (nine-point center) H (orthocenter) H (orthocenter) N (Nagel point) F (Furhman point) L (de Longchamps point) O (circumcenter) Table 9.
Correspondence table between points in the Euler ad base trianglesAccording to the above table we have buil the correspondence between points of the Darbouxcubic for the Euler triangle and triangle centers of the base triange.
Theorem 7.
New cubic 7 passes through circumcenter, orthocenter, Euler center, midpoint ofthe incenter and the orthocenter, vertices of the Euler triangle and middles of the triangle sides.
Finally, we observe correspondence of traingle centers between mid-arc and base trinalge.
TRIANGLE CONICS AND CUBICS
Darboux cubic in the Euler triangle New cubic for the base triangle
A, B, C (vertices) E , E , E (vertices of the Euler triangle) A (cid:48) , B (cid:48) , C (cid:48) (antipods of the triangle vertices) M , M , M (middles of the triangle sides) I (incenter) M IH (midpoint of incenter and orthocenter) O (circumcenter) E (nine-point center) H (orthocenter) H (orthocenter) L (de Longchamps point) O (circumcenter) Table 10.
Mtaching points for the Darboux cubic and Euler triangle
Figure 7.
New cubic based on Darboux cubic in the Euler trianle
Points for the mid-arc triangle Points for the base triangle O (circumcenter) O (circumcenter) H (orthocenter) I (incenter) Sy (Lemoine center) M MiI (midpoint of mittenpunktand incenter) L (de Longchamps point) Be (Bevan point) K (Kosnita point) S (Schiffler point) Table 11.
Correspondence table between points in the mid-arc and base trianglesBased on the correspondence between point sof the mid-arc and base triangles we discoverednew cubic which is based on Jerabek hyperbola.
Jerabek hyperbola for mid-arc triangle New hyperbola for the base triangle
A, B, C (vertices) A , A , A (middles of the arcs ofthe circumcircle) O (circumcenter) O (circumcenter) H (orthocenter) I (incenter) Sy (Lemoine point) M MiI (midpoint of mittenpunktand inenter) K (Kosnita point) S (Schiffler point) L (cid:48) (isogonal conjugate of the deLongchamps point) Be (cid:48) (isogonal conjugate of the Be-van point) Table 12.
Mtaching points for the Jerabek hyperbola and Euler triangle
RIANGLE CONICS AND CUBICS 9
Theorem 8.
New conic 8 is rectangular and passes through circumcenter, incenter, midpointof mittenpunkt and incenter, Schiffler point, and isogonaly conjugate point to the Bevan point.
Figure 8.
New cubic based on the Jerabek hyperbola for the mid-arc triangleTherefore, during the research of the triangle curves were derived three new triangle conicsnad five new triangle cubics. This is a significant result and leaves room for new investigations.Since, geometry of conic sections and other triangle curves are broadly used in the projectivegeometry 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 joinedby line segments in any order to form a hexagon, then the three pairs of opposite sides of thehexagon meet at three points which lie on a straight line.Let‘s consider the first derived triangle curve based on Jerabek hyperbola for the excentraltriangle. New hyperbola passes though excenters, Bevan point, incenter, mittenpunkt, deLongchaps point. We bulit a hexagon with verices in the given triangle centers and applyPascal’s theorem.Let I , I be excenters, and Be , M i , L be Bevan point, M i mittenpunkt, de Longchampspoint, respectively. We get the following results:
Corollary 9.
Concurent points of I Be and LI , BeM i and I L , M iI and I I belong to oneline. Corollary 10.
Concurent points of segments I M i and
BeI , BeL and II , M iL and II lieon one line. Simiraly, we have applied the same idea for the hexagon inscribed in the new hyperboladerived from the Jerabek hyperbola for the mid-arc triangle.Let A , A 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 Be and SI , IA and OA , BeA and OS belong to one line. Corollary 12.
Points of intersection of line segments
BeO and A A , BeS and IA , A S and IA belong to a straight line. Moreover, combination of two of the discovered triangle cubics gives us very interestingcorollary as well. Let’s consider new cubic derived from the Darboux cubic for the excentraltriangle 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, orthocen-ter, nine-point center, circumcenter, let’s name it P ( x, y ). The second mentioned new cubicpasses 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 com-mon points which are nine-point center, circumcenter and orthocenter. Moreover, this threepoints belong to Euler line, let it has an equation ax + by + c = 0. Since, we have two cubicswhich pass through points which belong to one line, there exists such integer t such that thefollowing holds: P ( x, y ) − tQ ( x, y ) ... ax + by + c . Therefore, Euler line is linear component ofthe composition of two new cubics. In addition, points of intersection of the linear componentwith the curve are inflection points [4]. Corollary 13.
Euler line is a linear component of the composition of new triangle cubic (passesthrough bases of altitudes, vertices, orthocenter, nine-point center, circumcenter) and new tri-angle cubic (passes through middles of the triangle sides, Speaker point, nine-point center,circumcenter, orthocenter). Moreover, orthocenter, circumcenter, and nine-point center areinflection point of the composition of these two curves.
The above corollaries prove that the discovered in the research new triangle curves could beapplied in different geometric areas and studied in advanced.
Remark . A further continue of our research consists in the same analysis of singularities asprovided 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 newtriangle cubics, what is very significant result for the classical geometry. In addition, was shownthat proceedings of the study could be applied not only in Euclidian space, but in projectiveas well. However, the main result of the research was developed algorithm of deriving newtriangle curves. This opens an opportunity for creating more triangle curves, while applyingthe method for various triangles, points, and geometric constructions.The developed idea significantly simplifies the question of creating curves passing throughtriangular centers. However, it opens up a number of new questions. Which interesting prop-erties do new curves have? What is the topological nature of these transformations? Is itpossible to apply a similar idea to non-Euclidean objects? Could one use the same method overan arbitrary finite field? Can this idea be further generalized?
References [1] Clark Kimberling.
Triangle centers and central triangles , 1998, Utilitas Mathematica.[2] N.J. Wildberger. Neuberg cubics over finite fields.
Algebraic Geometry and its Applications , volume 5,488-504, 2008.[3] H.M. Cundy and C.F. Parry. Some cubic curves associated with triangle.
Journal of Geometry , 53, 41-66,2000[4] Robert J. Walker.
Algebraic Curves , 1978, Springer-Verlag New York.[5] Pinkernell, G. M. ”Cubic Curves in the Triangle Plane.” J. Geom. 55, 141-161, 1996.[6] Drozd, Y. A., R. V. Skuratovskii.