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Let P be a point in the plane of the reference triangle ABC and denote by
The following tables give the loci of P such that two of these triangles are perspective. Note the frequent occurence of the Darboux cubic K004. true means the triangles are perspective for any P, true(H) means the triangles are actually homothetic L denotes the line at infinity, C denotes the circumcircle, 6B the union of the six bisectors. |
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Loci related to refP perspective with a fixed triangle T This section written with Martin Acosta's cooperation In this section we study the locus L(P) of P such that refP is perspective with a FIXED given triangle T and, if possible, the locus L(Q) of the perspector Q. Several loci are already available in the table above when T is the reference triangle ABC.
Let T = A1B1C1. Denote by A2, B2, C2 the reflections of A1 in BC, B1 in CA, C1 in AB respectively. Let A3 = BC2 /\ CB2 and define B3, C3 similarly. The locus L(P) is a circular circum-cubic which contains M, A2, B2, C2, A3, B3, C3. L(P) also contains A1 if and only if M lies on the isogonal pK with pivot the midpoint of the altitude AH. Hence, L(P) contains the three points A1, B1, C1 if and only if M is an in/excenter. When M = I, the cubic is K269 = pK(X6, X515) and when M is an excenter, it is the isogonal pK whose pivot is the corresponding extraversion of X(515), the infinite point of the line IH. L(P) is a K0 (i.e. a cubic without term in xyz) if and only if M lies on the Napoleon cubic K005, and in this case, it is always a pK. This is the case of K269. ***
Let T = A1B1C1. Denote by A2, B2, C2 the reflections of A1 in BC, B1 in CA, C1 in AB respectively. Let A3 = BC2 /\ CB2 and define B3, C3 similarly. The locus L(P) is a circular circum-cubic which contains M* = isogonal conjugate of M, A2, B2, C2, A3, B3, C3. L(P) also contains A1 if and only if M lies on the isogonal pK with pivot the antipode of A on the circumcircle. Hence, L(P) contains the three points A1, B1, C1 if and only if M is an in/excenter. When M = I, the cubic is K269 = pK(X6, X515) and when M is an excenter, it is the isogonal pK whose pivot is the corresponding extraversion of X(515), the infinite point of the line IH. L(P) is a K0 (i.e. a cubic without term in xyz) if and only if M lies on K270 = pK(X6, X1503). |
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Loci related to circumperp triangles The circumperp triangles CP1, CP2 and their tangential triangles TCP1, TCP2 are defined in Clark Kimberling's TCCT (§§ 6.21 upto 6.23). The vertices of CP1 (resp. CP2) are the second intersections of the circumcircle with the external (resp. internal) bisectors of ABC. The following table gives the loci of P such that a triangle related to P is perspective with one of these four triangles. |
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These triangles CP2 and CP1 can be seen as the circumcevian and circumanticevian triangles of the incenter. This latter point may be replaced by any other fixed point Q as far as it does not lie on a sideline or on the circumcircle. The most interesting generalization is obtained with ccvQ, the circumcevian triangle of Q, since we always find three related pK. The following table gives the loci of P such that ccvQ is perspective to cevP, acvP (or cacP) and also the locus of the perspector which is the same for all these triangles. |
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Loci related to other triangles The following table gives another selection of loci related to several triangles as in Clark Kimberling TCCT p.155 & sq and other "classical" triangles. |
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Cevian and Anticevian, Pedal and Antipedal triangles of a fixed point Q |
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Note : for any Q and for any P on K004, pedP and acvQ are perspective and the locus of the perspector is a cubic which is the transform of the Lucas cubic K007 under the mapping M -> M/Q (M-Ceva conjugate of Q). This cubic is a pK in acvQ with pivot the isogonal of Q in the orthic triangle and isopivot X(69)/Q. When Q = H, it is also a pK in ABC. |
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Intangents and Extangents triangles details in TCCT §6.16 p.161 and §6.17 p.162 |
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Reflected triangles A'B'C' is the triangle formed by the reflections of A, B, C in the sidelines of ABC. The hexyl triangle is the triangle formed by the reflections of the excenters in the circumcenter O of ABC. A1B1C1 is the triangle formed by the reflections of O in the vertices of ABC (TCCT §6.13). |
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See other loci related to the four Brocard triangles. See also K585, K586 for cubics related to the Morley triangle. |
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Loci related to cevian and anticevian triangles The following table gathers together a good number of pKs, the loci of P whose cevian / anticevian triangle is perspective at Q to a certain triangle T. Note that the locus of the perspector Q is the same pK in both cases. This is a direct adaptation of César Lozada's tables in ETC, see X(8782) and X(8856), with some additional informations. |
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*** In the yellow line of the table above, the triangle T is the circumcevian triangle of X(1). More generally, if T is the circumcevian triangle MaMbMc of some fixed center M ≠ X(6), we find several analogous pKs. With M = X(6), these are degenerated, excluded in the sequel. More precisely, the loci of P whose cevian / anticevian triangle is perspective (at Q) to T are pK0 / pK1. Note that the locus of the perspector Q is the same pK in both cases, namely pK2. The isogonal transforms of pK0, pK1 and pK2 are denoted pK0*, pK1* and pK2*. Properties of these cubics |
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Additional properties pK0 pK1 meets (O) again at Q1, Q2, Q3 where the tangents concur at the symmedian point of the triangle, but this point is not on the cubic. These points also lie on pK(X32, M) and on pK(X6 x M, X6). The inconic (of ABC) with perspector P1 is inscribed in Q1Q2Q3 and the cubic passes through its six contacts with the sidelines of both triangles. pK1 meets (L∞) at three points which lie on pK(ctP1, X69) and on pK(X3, a(X4 x ctP1). pK2 meets (O) again at Ma, Mb, Mc where the tangents concur at X(6) x a(gM)^2) on the curve. The remaining points on the sidelines of T are the vertices Na, Nb, Nc of a triangle perspectif at gcgM to MaMbMc and at P1 ÷ (agM)^2 to ABC, these two points lie on the cubic. pK0* pK1* pK2* is a central cubic with center cgM *** The table below lists a large number of these cubics. |
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The pink cells contain points on the Lemoine axis. pK0*and pK1* are members of CL007 and CL048 respectively. They are isotomic transforms of each other and both pass through the infinite points of the Steiner ellipse.
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