INTERNATIONAL JOURNAL OF GEOMETRY Vol. 4 (2015), No. 2, 5 - 25 GENERALISATIONS OF THE PROPERTIES OF THE NEUBERG CUBIC TO THE EULER PENCIL OF ISOPIVOTAL CUBICS Ivan Zelich and Xuming Liang Abstract. This paper discusses results that arise in specific configu- rations pertaining to invariance under isoconjugation. The results lead to crucial theorems in both Euclidean and Projective geometry. After dis- cussion of important theorems and properties of associated configurations, the authors present and prove a new result and its application in difficult geometrical configurations. 0. Introduction While locus { a set of points satisfying some condition { is widely used in various mathematical areas, its geometric nature makes it practically in- separable from the field of Geometry. The simplest loci like lines and circles are the building blocks of Euclidean Geometry originating more than two thousand years ago. Recently, conical loci such as ellipses, parabolas and hy- perbolas were incorporated into the picture, and since then both Euclidean and Projective Geometry have combined giving a completely new perspec- tive on Geometry as a whole. It is in human nature, however, to consider what other loci there may be, and this paper turns to an interesting type of locus, the cubic. Similar to conics which are second degree curves, cubics are simply third degree curves. Generally, cubics are very complex and hard to deal with in a Projective light, but cubics that are intrinsically projective give rise to amazing results. Indeed, this paper is interested in the pencil of isopivotal cubics having pivot on the Euler line of a fixed triangle. While many synthetic results have been formed on these types of cubics [6], the authors have discovered new results that generalize a well-known locus { the Neuberg cubic { to the Euler pencil of isopivotal cubics. ||||||||||||{ Keywords and phrases: Neuberg Generalisation, Euler Pencil, isopiv- otal cubics, pedal triangle, Sondat's Theorem (2010)Mathematics Subject Classification: 51M04 Received: 1.31.2015. In revised form: 20.08.2015. Accepted: 10.09.2015. 6 Ivan Zelich and Xuming Liang The major result discovered can be stated as follows: Theorem 0.1 (Liang-Zelich). Consider a point on an isopivotal cubic with pivot on the Euler line of a given triangle. Then this point lies on the same isopivotal cubic constructed in its pedal triangle. In this context, the term, \same," will refer to an isopivotal cubic H hav- ing pivot cutting the Euler line in a fixed directed ratio w.r.t. an arbitrary triangle. Explicitly, for this section, we will call T being the pivot of H and TO t being the directed ratio TH where O; H are the circumcenter and ortho- center of the arbitrary triangle. This result could be seen as a generalization of a well-known fact that if a point P lies on the Neuberg cubic, then P lies on the Neuberg cubic of its pedal triangle [6]. This fact is not known for other well-known cubics, so proving such a property is very difficult even with the aid of coordinates. Thus it is necessary to find more properties of the configuration in Theorem 0:1. Indeed, a remarkable generalization is given in Definition 2:3 of the prop- erty that for a point P on the Neuberg cubic of an arbitrary triangle ABC, the triangle formed by the circumcenters of ⊙P BC; ⊙P AC; ⊙P AB (the `Carnot' triangle of P ) is perspective with ABC [8]. A rephrase of Defini- tion 2.3 can be stated as follows: Theorem 0.2 (Generalized Carnot triangles). Consider the image of Carnot 1 triangle of P dilated with factor t about P . This resulting triangle is per- spective with ABC if and only if P 2 H. Since the Carnot triangle of P is similar to the pedal triangle of its isog- onal conjugate w.r.t. ABC, by definition an isopivotal cubic is invariant under isoconjugation, and Assertion 3 in section 2, we can conclude the aforementioned generalization implies the following: Theorem 0.3 (Generalized Pedal triangles [7]). Suppose the reflection tri- angle of a point P 2 H is dilated about P with ratio t. The resulting triangle is perspective with ABC. 1 This result is known for t = 1; 2 , which are the Neuberg, Napolean- Feurerbach, Darboux and Thomson cubic respectively [6]. This generaliza- tion was published in the Journal of Geometry by Guido Pinkernell [7]. The last generalized property given in this paper has only been known for the Neuberg cubic [4]. This property is further generalized in Theorem 2:5, but a sub-case of this Theorem can be stated as follows: Theorem 0.4 (Generalized Euler lines). Define a (x; P )-Euler line of a triangle XYZ as the line through the orthocenter of XYZ and the image of the circumcenter under dilation x about P . The (t; P )-Euler lines of BP C; AP C; AP B concur on the (t; P )-Euler line of ABC if and only if P 2 H. Generalized properties of the Euler pencil of isopivotal cubics 7 All these theorems are corollaries of the properties synthetically proven in section 2. In section 3, we provide some amazing applications that demon- strate the sheer strength of the Liang-Zelich Theorem and associated con- figurations. 1. Notations, definitions and auxiliary results 1.1. Notations and Definitions. Given a triangle ABC, O; H will always be used to denote its circumcenter and orthocenter respectively. When P; Q are used in the same context, they will always be isogonal conjugates with respect to (w.r.t. in short) ABC unless stated otherwise (e.g. Theorem 1.5). The authors will refer to OXYZ (ABC), when appropriate, as the con- currence of the perpendiculars from X to BC, Y to AC and Z to AB for orthologic triangles XYZ and ABC. Given a point P and triangle ABC, let OPA;OPB;OPC denote the cir- cumcenters of P BC; P AC; P AB. We will call OPAOPBOPC the P -Carnot triangle w.r.t. ABC. Similarly, OQAOQBOQC is the Q-Carnot triangle w.r.t. ABC. Note that these notations will be heavily used, especially in section 2. 1.2. Auxiliary Results. In this section, the authors prove some auxiliary results that will be used throughout this paper. Proposition 1.1. Consider an arbitrary triangle ABC on a rectangular hyperbola. Then the orthocenter of ABC is also on the rectangular hyperbola. The converse is also true. Proof. Under isogonal conjugation w.r.t. ABC, suppose the hyperbola maps to a line `. Let ` meet ⊙ABC at two points X; Y . Since both X and Y go to infinity under isogonal conjugation, it follows that their isogonal con- jugates are in directions that are mutually perpendicular. It is well-known that this happens if and only if X; Y are antipodal, thus the circumcenter O of ABC is on `. Hence, under isogonal conjugation, the orthocenter of ABC is on the rectangular hyperbola. Theorem 1.2 (Sondat's Theorem). Consider two triangles ABC and A0B0C0 0 0 0 0 that are mutually orthologic with orthology centers P; P (OABC (A B C ) = 0 0 P; OA0B0C0 (ABC) = P ) and perspective with perspector Q. Then, Q 2 PP . Lemma 1.3. Let ABC and A0B0C0 be two fixed triangles that are not homothetic. Let P be a point such that the parallels through A0;B0;C0 to AP; BP; CP respectively concur at a point Q. Then the locus of points P is either a circumconic of ABC or the line at infinity. Proof. Suppose the parallels to A0B0;A0C0 through B; C respectively meet at `A; define `B; `C respectively. Note that: B`A \ A`B; B`C \ C`B; C`A \ A`C ; all lie on the line at infinity. So by the converse of Pascal's Theorem on hexagon A`BC`AB`C , it follows that A; B; C; `A; `B; `C all lie on a conic C. 8 Ivan Zelich and Xuming Liang A Y H O B C X Figure 1. A fundamental result on rectangular hyperbolas. Q A' P' A B' C' P C B Figure 2. Sondat's Theorem - a little-known theorem. C0 0 0 0 0 0 0 Define an equivalent conic w.r.t. A B C and points `A; `B; `C . But, by Pascal's Theorem on the hexagon AAC`AB`C we have that if: ∗ ∗ AA \ B`A = A ; AC \ B`C = C ; ∗ ∗ 0 0 0 ∗ k 0 0 k 0 0 then A BC is homothetic to A B C . Since AC `AC and AB `AB , 0 the points A and `A are corresponding under the aforementioned homothety. Generalized properties of the Euler pencil of isopivotal cubics 9 C ∗ 0 0 Hence the tangent at A to , which is equivalent to AA is parallel to A `A. Now, take a point P on C. Let the point at infinity on AP be 'A: define 0 0 0 0 'B;'C similarly. Then, A(P; `A; `B; `C ) = A ('A;A ; B ;C ). Hence: 0 0 0 0 0 0 0 0 0 0 0 0 A ('A;A ; B ;C ) = B ('B;A ;B ;C ) = C ('C ;A ;B C ): 0 0 0 0 Therefore, A 'A;B 'B;C 'C concur at a point Q 2 C . Hence, we have proven that all points P 2 C satisfy the property in this Lemma, so it suf- fices to prove uniqueness of this locus apart from the line at infinity. 0 0 0 0 Consider a point P moving on AP and define 'A;'B;'C to be the points at infinity on AP 0; BP 0;CP 0 respectively.