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Let P, Q be two distinct points with cevian triangles PaPbPc, QaQbQc respectively. Denote by Oa, Ob, Oc the circumcenters of triangles APaQa, BPbQb, CPcQc.

When P is the orthocenter H of ABC, these points Oa, Ob, Oc are clearly the midpoints of AQa, BQb, CQc hence the triangles ABC and OaObOc are perspective at Q for all Q.

In the sequel, we suppose that P ≠ H is a fixed point and denote by K(P) the locus of Q such that ABC and OaObOc are perspective and by K'(P) the locus of the perspector. From the remark above, K(P) must contain H and K'(P) must contain P.

When Q is the isogonal conjugate P* of P, the circles APaQa, BPbQb, CPcQc are tangent at A, B, C respectively to the circumcircle (O) of ABC hence ABC and OaObOc are perspective at O on K'(P) then P* must be on K(P).

Main theorem

For P ≠ H, the locus of Q such that the triangles ABC and OaObOc are perspective is a focal circum-cubic K(P) passing through H and P*.The locus of the perspector is also a focal circum-cubic K'(P), passing through O and P, which is the isogonal transform of K(P).

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Properties of K(P)

• The singular focus F of K(P) is the antigonal of P i.e. the antipode of P in the rectangular circum-hyperbola H(P) passing through P. The center of H(P) is denoted Ω. Hence, if the singular focus of K(P) is F ≠ H, then the singular focus of K(F) is P and each cubic is the antigonal transform of the other. Furthermore, the singular focus of K'(P) is the inverse of F in (O).

• K(P) passes through P1, the midpoint of H and aP, where aP is the anticomplement of P.

• The line passing through aΩ and P1 meets (O) at aΩ and another point S which lies on K(P).

• The line FP1 meets H(P )again at P2 which lies on K(P).

• The real asymptote A(P) of K(P) is parallel to the line OP* hence K(P) passes through the infinite point of OP*.

• The parallel at H to A(P) meets K(P) again at P3 on the lines FP* and SP1.

• The line OP* above meets K(P) again at P4 on the line HF.

• The parallel at P to A(P) meets K(P) at P2 and another point P5 on the line SP4.

• The line HP1 meets H(P )again at P6 on the line P*P5.

• P7 = P1P4 /\ P3P5 is another point on K(P).

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Special cases

• When P lies on the line at infinity (L∞), K(P) must split into (L∞) and the rectangular circum-hyperbola which is the isogonal transform of the line OP.

• When P lies on (O), the center Ω of H(P) lies on the nine point circle and then F = H. In this case, K(P) contains X(265) and the reflection of P in the Euler line (E) of ABC. The real asymptote passes through O.

• K(X265) splits into (O) and (E). Hence, for any P ≠ H on (E), K(P) passes through X(265) and F lies on K025 which is K(X3) with singular focus X(265).

• K(P) passes through P (and then K'(P) passes through P*) if and only if P lies on Q038, a circular quintic passing through X(i) for i in {1, 4, 5, 80, 1113, 1114, 1263, 2009, 2010}.

• K(P) passes through O (and then K'(P) passes through H) if and only if P lies on K025, a strophoid passing through X(i) for i in {4, 30, 265, 316, 671, 1263, 1300, 5080, 5134, 5203, 5523, 5962, 10152, 11604, 11605, 11703, 13495, 16172, 19552, 31862, 31863, 34150, 34169, 34170, 34171, 34172, 34173, 34174, 34175, 34239, 34240, 37888}.

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Special types of K(P)

• K(P)is a K0 (no term in x y z) if and only if P lies on K337 = pK(X232, X4) passing through X(i) for i in {4, 114, 371, 372, 511, 2009, 2010, 3563, 5000, 5001, 32618, 32619}.

• K(P)is a nK if and only if P lies on the circum-conic with perspector X(216) passing through X(110), X(265), X(1625). In this case, the pole lies on the circum-conic with perspector X(51), the root lies on a complicated quartic passing through X(2), X(648), X(18883). The isogonal conjugate of the singular focus lies on the circle with diameter OH. For example, K(X110) = nK(X1989, X18883, X4) is the isogonal transform of K933.

• K(P)is a psK if and only if P lies on a complicated tricircular nonic passing through H and the feet Ha, Hb, Hc of the altitudes.

• In particular, K(Ha), K(Hb), K(Hc) are three focal pKs with singular foci Ha, Hb, Hc respectively.

 

The following table gives a selection of these focal cubics K(P) and K'(P), with singular foci F and F', for P ≠H not lying on the line at infinity.

K"(P), with singular focus F", is the inverse of K'(P) in the circumcircle and also the cubic K(F) where F = antigonal(P) is the singular focus of K(P).

P

K(P)

F

K'(P)

F'

K"(P)

F"

remarks

X(1)

K529

X(80)

 

X(10260)

 

X(36)

 

X(2)

K300

X(671)

 

X(11643)

K302

X(23)

 

X(3)

K025

X(265)

K039

X(5961)

(L∞) U (J)

 

K(P) and K'(P) are strophoids

X(5)

K464

X(1263)

K467

X(14367)

K466

X(2070)

 

X(6)

K477

X(67)

 

X(3455)

 

X(187)

 

X(13)

K1132b

X(14)

K1133b

X(6105)

K1133a

X(6104)

 

X(14)

K1132a

X(13)

K1133a

X(6104)

K1133b

X(6105)

 

X(23)

K481

X(316)

 

X(21395)

K473

X(2)

 

X(67)

K298

X(6)

 

X(187)

 

X(3455)

 

X(69)

 

X(895)

K698

X(6091)

 

X(5866)

 

X(74)

K530

X(4)

 

X(186)

K187

X(74)

K(P) is a central cubic

X(80)

K681

X(1)

 

X(36)

 

X10260)

 

X(98)

 

X(4)

 

X(186)

K433

X(98)

 

X(110)

 

X(4)

K933

X(186)

K932

X(110)

K(P) and K'(P) are nKs

X(265)

(O) U (E)

X(3)

(L∞) U (J)

 

K039

X(5961)

(J) is the Jerabek hyperbola

X(316)

K302

X(23)

K473

X(2)

 

X21395)

 

X(671)

K473

X(2)

K302

X(23)

 

X(11643)

 

X(842)

K301

X(4)

 

X(186)

K072

X(842)

K"(P) is a nK

X(895)

 

X(69)

 

X(5866)

K698

X(6091)

 

X(953)

K275

X(4)

K274

X(186)

K165

X(953)

K(P) and K'(P) are strophoids

X(1263)

K465

X(5)

K466

X(2070)

K467

X(14367)

K(P) is a central cubic

X(2070)

 

X(19552)

 

X(21394)

K465

X(5)

 

X(2698)

 

X(4)

 

X(186)

K166

X(2698)

 

X(3563)

 

X(4)

 

X(186)

K164

X(3563)

 

X(19552)

K466

X(2070)

K465

X(5)

 

X(21394)

 

The lines in yellow are those with P on (O) hence the singular foci of K(P) and K'(P) are X(4) and X(186) respectively.

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Additional cubics (computed by Peter Moses)

cubic

X(i) on the cubic for i =

K'(X2) = K''(X671)

2, 3, 67, 98, 182, 186, 187, 3431, 7607, 7771, 11643, 30542

K(X1337)

4, 13, 17, 622, 633, 3479, 8174, 11600, 34219, 36766

K'(X54) = K''(X33565)

3, 54, 74, 1154, 2070, 3520, 5944, 11273, 15620, 34418

K(X20)

4, 64, 265, 382, 2071, 3357, 6000, 10152, 15318, 33641

K'(X6) = K''(X67)

3, 6, 23, 74, 378, 511, 3425, 3455, 6800, 30541

K''(X2071) = K(X10152) = K'(X34170)

3, 4, 20, 2693, 6000, 11589, 11744, 23240, 34170

K(X2070)

4, 5, 265, 2888, 3153, 11584, 19552, 32423, 33565

K'(X2070) = K''(X19552)

3, 54, 186, 2070, 3432, 14367, 14979, 21394, 34418

K(X1338)

4, 14, 18, 621, 634, 3480, 8175, 11601, 34220

K'(X842)

3, 23, 186, 187, 249, 511, 842, 3447, 22259

K(X622)

4, 14, 15, 61, 627, 1337, 3439, 5615, 11602

K''(X622) = K'(X1337)

3, 15, 61, 1337, 3439, 3442, 6104, 8471, 32627

K(X621)

4, 13, 16, 62, 628, 1338, 3438, 5611, 11603

K''(X621) = K'(X1338)

3, 16, 62, 1338, 3438, 3443, 6105, 8479, 32628

K(X104)

4, 65, 265, 517, 1320, 2687, 5080, 10742, 17101

K(X98)

4, 6, 265, 316, 511, 842, 1916, 6033, 37841

K'(X23) = K''(X316)

3, 6, 23, 186, 842, 3455, 11649, 12584, 21395

K''(X21) = K(X5080) = K'(X11604)

3, 4, 65, 104, 1325, 2771, 11604, 22765, 34442

K''(X6) = K'(X67)

2, 3, 67, 74, 187, 542, 10295, 11653, 14907

K'(X5080) = K(X11604)

3, 4, 21, 517, 2687, 5080, 5172, 10693

K(X2071)

4, 20, 265, 2777, 11744, 22802, 31726, 34170

K'(X622) = K''(X1337)

3, 13, 16, 17, 622, 3479, 3489, 14144

K'(X621) = K''(X1338)

3, 14, 15, 18, 621, 3480, 3490, 14145

K'(X104)

3, 21, 104, 186, 1319, 2771, 17100, 34442

K''(X104)

3, 4, 8, 56, 104, 517, 1325, 10693

K'(X98)

2, 3, 98, 186, 542, 1691, 3455, 5152

K(X54)

4, 5, 30, 1141, 3521, 6288, 19552, 33565

K''(X54) = K'(X33565)

3, 5, 74, 1157, 13619, 32423, 33565, 35888

K(X21)

4, 65, 265, 517, 1325, 1389, 11604, 37230

K'(X20) = K''(X10152)

3, 20, 186, 1294, 6759, 11270, 11589, 11744

K''(X20) = K'(X10152) = K(X34170)

3, 4, 64, 1294, 2071, 2777, 6760, 10152