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K631

∑ (b+c-a) x^2 (c^2 y - b^2 z) = 0

or

∑ a^2 [(c+a-b)y - (a+b-c)z] yz = 0

X(6), X(7), X(509), X(1486), X(7133), X(42013), X(53134), X(53135)

A'B'C' = cevian triangle of X(7) = intouch triangle

K631 = pK(X56, X7) is the only Poncelet poristic cubic which is a pivotal cubic passing through the vertices of the intouch triangle. See Pseudo-Pivotal Cubics and Poristic Triangles, §2.3.

See the related cubics K360, K577, K578. which are psK(X56, X7, M) for M = X(1), X(2), X(513) respectively, with the same property.

***

More generally, the cubic pK(Ω = p : q : r, P = u : v : w) meets (O) again at Q1, Q2, Q3 such that the incircle (of ABC) is inscribed in Q1Q2Q3 if and only if :

• Ω = X6 x P x a(X8 x P) = a^2 u ((a-b-c) u+(a-b+c) v+(a+b-c) w) : : .

• P = Ω ÷ X6 x a(Ω ÷ X56) = b^2 c^2 p (b^2 (a-b-c) c^2 p+a^2 c^2 (a-b+c) q+a^2 b^2 (a+b-c) r) : : .

where x and ÷ denote barycentric product and quotient respectively. Obviously, Ω = X(56) ⟺ P = X(7) as above.

Remarks :

• Ω and P coincide in X(1401).

• When Ω lies on the trilinear polar L(M) of a poinr M, P lies on the bicevian conic C(M x X76, X7).

In particular, with M = X(56), we find P on the incircle. L(M) passes through {649, 854, 1404, 3063, 3310, 4832, 6371, 7180, 14412, 17424, 20981, 21758, 22383, 23472, 43924, 50492, 52635, 52970, 53539, 54275, 57181}

• When P lies on the trilinear polar L(M) of a poinr M, Ω lies on the bicevian conic C(M x X6, X56).

In particular, with M = X(7), we find Ω on the inconic with perspector X(56) and L(M) is the Gergonne line passing through {241, 514, 650, 665, 905, 1323, 1375, 1465, 1638, 3002, 3004, 3008, 3015, 3669, 3676, 3776, 3911, 3960, 4369, 4763, 4841, 5723, 6357, 7178, 7180, 7181, 7214, 7658, 8074, etc}.

***

Lists of pairs {Ω, P}, {P, Ω} and a selection of corresponding pK(Ω, P) by Peter Moses

{Ω, P} : {6, 145}, {25, 196}, {31, 57}, {32, 1486}, {36, 41873}, {41, 3174}, {42, 3175}, {48, 224}, {55, 8055}, {56, 7}, {57, 30545}, {101, 30721}, {181, 56326}, {184, 3173}, {187, 16597}, {213, 22277}, {237, 16591}, {244, 4077}, {512, 8287}, {604, 1}, {608, 44696}, {647, 34846}, {649, 11}, {663, 26932}, {667, 1086}, {669, 16592}, {902, 16594}, {978, 75}, {1015, 513}, {1042, 1439}, {1055, 10427}, {1193, 39774}, {1201, 1122}, {1333, 229}, {1357, 3676}, {1397, 222}, {1400, 65}, {1401, 1401}, {1402, 226}, {1404, 1317}, {1405, 16236}, {1407, 9533}, {1410, 3668}, {1415, 40577}, {1423, 85}, {1458, 39775}, {1464, 41801}, {1468, 39773}, {1475, 15185}, {1946, 16596}, {1977, 50514}, {2175, 218}, {2178, 10940}, {2183, 39776}, {2187, 54414}, {2212, 20613}, {2223, 16593}, {2260, 39772}, {3009, 20343}, {3052, 2}, {3063, 3022}, {3122, 23755}, {3124, 12072}, {3209, 4}, {3248, 48334}, {3250, 55061}, {3271, 514}, {3310, 3326}, {3747, 46842}, {4017, 1111}, {4832, 31890}, {7083, 7195}, {7113, 39778}, {7117, 522}, {7180, 1365}, {8632, 38989}, {8641, 13609}, {8642, 40615}, {8643, 40617}, {8645, 40629}, {8648, 46398}, {16502, 43916}, {16945, 47636}, {17053, 17114}, {17082, 6063}, {18753, 56707}, {20467, 726}, {20662, 518}, {20972, 519}, {20981, 3023}, {20982, 523}, {21748, 39783}, {21755, 512}, {21758, 3025}, {22383, 1364}, {34068, 15727}, {38363, 918}, {39201, 16595}, {40956, 55010}, {41280, 7251}, {42461, 69}, {43924, 1358}, {51329, 9436}, {51641, 53538}, {52411, 56}, {52635, 1362}, {53538, 23599}, {53539, 3323}, {55054, 830}, {57181, 1357}

{P, Ω} : {1, 604}, {2, 3052}, {4, 3209}, {7, 56}, {11, 649}, {56, 52411}, {57, 31}, {65, 1400}, {69, 42461}, {75, 978}, {85, 1423}, {145, 6}, {196, 25}, {218, 2175}, {222, 1397}, {224, 48}, {226, 1402}, {229, 1333}, {512, 21755}, {513, 1015}, {514, 3271}, {518, 20662}, {519, 20972}, {522, 7117}, {523, 20982}, {726, 20467}, {830, 55054}, {918, 38363}, {1086, 667}, {1111, 4017}, {1122, 1201}, {1317, 1404}, {1357, 57181}, {1358, 43924}, {1362, 52635}, {1364, 22383}, {1365, 7180}, {1401, 1401}, {1439, 1042}, {1486, 32}, {3022, 3063}, {3023, 20981}, {3025, 21758}, {3173, 184}, {3174, 41}, {3175, 42}, {3323, 53539}, {3326, 3310}, {3668, 1410}, {3676, 1357}, {4077, 244}, {6063, 17082}, {7195, 7083}, {7251, 41280}, {8055, 55}, {8287, 512}, {9436, 51329}, {9533, 1407}, {10427, 1055}, {10940, 2178}, {12072, 3124}, {13609, 8641}, {15185, 1475}, {15727, 34068}, {16236, 1405}, {16591, 237}, {16592, 669}, {16593, 2223}, {16594, 902}, {16595, 39201}, {16596, 1946}, {16597, 187}, {17114, 17053}, {20343, 3009}, {20613, 2212}, {22277, 213}, {23599, 53538}, {23755, 3122}, {26932, 663}, {30545, 57}, {30721, 101}, {31890, 4832}, {34846, 647}, {38989, 8632}, {39772, 2260}, {39773, 1468}, {39774, 1193}, {39775, 1458}, {39776, 2183}, {39778, 7113}, {39783, 21748}, {40577, 1415}, {40615, 8642}, {40617, 8643}, {40629, 8645}, {41801, 1464}, {41873, 36}, {43916, 16502}, {44696, 608}, {46398, 8648}, {46842, 3747}, {47636, 16945}, {48334, 3248}, {50514, 1977}, {53538, 51641}, {54414, 2187}, {55010, 40956}, {55061, 3250}, {56326, 181}, {56707, 18753}

{Ω, P}

pK(Ω, P)

X(i) on pK(Ω, P) for i =

{6, 145}

K1086

{1, 145, 2136, 2137, 3445}

{25, 196}

 

{196, 2192, 20034}

{31, 57}

 

{55, 57, 365}

{32, 1486}

 

{6, 1486, 3433, 34121, 34125, 60557}

{56, 7}

K631

{6, 7, 509, 1486, 7133, 42013, 53134, 53135}

{57, 30545}

 

{174, 2162, 30545}

{181, 56326}

 

{40148, 56326, 60537}

{604, 1}

K632

{1, 56, 84, 221, 266, 2067, 2362, 6502, 16232, 30335, 30336, 53116, 53117}

{608, 44696}

 

{64, 84, 207, 208, 3345, 44696}

{1333, 229}

 

{229, 267, 501, 34435}

{1397, 222}

 

{25, 222, 60538}

{1400, 65}

 

{1, 65, 1046, 60543}

{1407, 9533}

 

{57, 8917, 9533, 11051}

{1415, 40577}

 

{59, 513, 3446, 38863, 40577}

{1977, 50514}

 

{649, 14088, 50514}

{2175, 218}

 

{218, 60530}

{3124, 12072}

 

{661, 12072, 60551}

{3209, 4}

 

{4, 221, 2362, 16232}

{3271, 514}

 

{514, 663, 26721, 60536}

{7117, 522}

 

{513, 521, 522, 1459, 6129, 59973}

{51329, 9436}

 

{672, 1447, 1914, 9436}

{52411, 56}

 

{3, 56, 2067, 6502, 15375, 42461}

{?, 8}

 

{8, 1616, 30556, 30557}

{?, 9}

 

{9, 2066, 5414, 21002}

{?, 10}

 

{1, 10, 191, 3214, 3915}

{?, 34}

 

{34, 56, 1406, 3556, 36103}

{?, 4017}

 

{513, 3733, 4017, 43924}

{?, 21127}

 

{513, 657, 21127, 60547}

{?, 24002}

 

{513, 21007, 24002, 57167}

{?, 57792}

 

{7, 21778, 31604, 37800, 57792}

pK(Ω, P) meeting (L∞) and (O) as an isogonal pK (yellow lines)

The points on (O) are those of K1086 = pK(X6, X145) hence the triangle Q1Q2Q3 has orthocenter X(145).

• Ω on psK(X41280 = X32 x X56, X56, X6), nodal cubic with node X(604), passing through {6, 604, 2175, 3209, 52411}.

• P on K360 = psK(X56, X7, X1) = spK(X8, X1), a Lemoine generalized cubic with node X(1), passing through {1, 4, 56, 145, 218, 279, 1433, 14260, 14584, 52382, 56634, 56635, 56636, 56637, 56638, 56639, 56640, 56641, 56642, 56643, 56644, 56645, 56646, 56647, 56648, 56840}.

• isopivot Q on a nodal circum-cubic with node X(56) passing through {3, 56, 221, 3207} and the points on (O) of K1086.