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K1367

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X(2), X(4), X(6), X(115), X(125), X(338), X(523), X(1640), X(14086), X(14223), X(36189), X(41254), X(51480), X(60500)

X(65608) → X(65624)

vertices of the cevian triangle A1B1C1 of X(41254)

vertices of the anticevian triangle A'B'C' of X(523) = Schroeter triangle, see ETC X(8286). A', B', C' and X(14086) are the square roots of X(115), X(14086) being that interior to ABC.

Geometric properties :

K1367 is another example of pK with pole X(115). See also K237, K238, K239, K672, K877, K878, K1144, K1378 and the table below computed by Peter Moses.

It is the only pK with pole X(115) through the points X(2), X(4), X(6) and then also through their X(115)-isoconjugates X(115), X(125), X(338).

The barycentric prooduct of K1367 by X(99) is K1368.

***

A selection of pK(X115, P) through at least 9 centers.

Recall that every cubic passes through the eight fixed points A, B, C, X(523), X(14086), A', B', C'.

P

X(i) on pK(X115, P) for i =

74

4, 30, 74, 125, 523, 3134, 12079, 14086, 15328, 43917, 52010, 55121, 58261

99

2, 99, 115, 523, 8029, 12076, 14086, 31644, 36953

111

2, 111, 115, 468, 523, 3143, 14086, 22105, 47138, 51258, 52628, 60040

141

141, 523, 594, 1086, 14086, 34294, 39022, 39023, 64649

148

2, 115, 148, 523, 4590, 13187, 14086, 31372, 61339

265

5, 30, 265, 523, 526, 8901, 10412, 12079, 14086, 35235

290

6, 290, 325, 338, 523, 14086, 35364, 44114, 51441, 60514

297

230, 297, 523, 525, 1312, 1313, 1503, 2501, 14086, 51404, 62592, 62593

1300

3, 265, 403, 523, 1300, 2970, 14086, 15328, 35235, 44665, 55121

1989

395, 396, 523, 1989, 14086, 23283, 23284, 38393, 62551

2373

25, 339, 523, 858, 2373, 8791, 14086, 47138, 60040, 62563

3239

523, 661, 1577, 3239, 4064, 6587, 7649, 14086, 58759

3569

523, 525, 804, 2501, 3569, 6130, 14086, 43665, 60036

3580

30, 523, 525, 2501, 3580, 12079, 14086, 16310, 39022, 39023

3766

512, 523, 661, 850, 918, 1577, 3766, 3837, 14086

3936

523, 661, 1577, 3936, 3943, 14086, 17757, 39022, 39023

4391

513, 523, 525, 661, 1577, 2501, 4036, 4391, 6588, 14086, 57185

9140

13, 14, 30, 523, 690, 5466, 9140, 12079, 14086, 30465, 30468

9979

523, 525, 690, 2501, 5466, 9979, 14086, 18310, 47138, 60040

10630

523, 524, 5099, 10415, 10630, 14086, 64258, 65609, 65611

14644

4, 5, 125, 523, 3154, 8901, 14086, 14644, 15453

14837

523, 656, 3064, 6587, 14086, 14837, 24006, 57243, 58759

16081

6, 338, 523, 6530, 14086, 16081, 34212, 41172, 60527

17924

523, 525, 650, 656, 2501, 14086, 15313, 17924, 24006, 55232

18896

141, 325, 523, 882, 2086, 14086, 14295, 18896, 34294, 51441

30786

2, 115, 523, 858, 9178, 14086, 30786, 35522, 40347

31057

2, 115, 523, 4049, 4120, 10026, 11599, 14086, 31057

34087

76, 523, 3124, 9148, 14086, 30736, 34087, 52625, 60028

38955

65, 523, 3657, 14086, 17757, 38955, 42759, 51421, 52383

40814

6, 262, 338, 523, 2592, 2593, 14086, 40814, 45030

41079

523, 525, 526, 2433, 2501, 10412, 14086, 14566, 41079, 46425

44427

523, 525, 1637, 2394, 2501, 14086, 14582, 15328, 44427, 55121

48540

6, 67, 338, 523, 524, 1989, 1990, 14086, 15328, 47138, 48540, 55121, 60040, 62551, 64258, see K1379, K1380

50188

4, 125, 523, 525, 2501, 6103, 14086, 36201, 50188

53331

512, 523, 850, 2395, 2799, 14086, 53331, 53567, 60037

53375

76, 523, 690, 698, 1916, 3124, 5466, 14086, 25322, 53375

54262

523, 525, 2501, 14086, 21958, 23301, 25423, 52631, 54262

54395

2, 115, 523, 525, 542, 2501, 14086, 54395, 56403

60502

6, 338, 523, 525, 2493, 2501, 2781, 14086, 57482, 60502, 60590

 

 

Remarks :

• pK(X115, P) meets the line at infinity at X(523) and two other points which lie on the circum-conic with perspector the barycentric product of X(523) and the complement cP of P. When P lies on the Euler line, this conic is a rectangular hyperbola.

• pK(X115, P) passes through a given point M ≠ X523 (and obviouly through its X115-isoconjugate M') when P lies on the line MM'.

For instance, pK(X115, P) passes through X(2) and X(115) when P lies on the line {2, 99, 111, 115, 126, 148, 543, 574, 620, 671, 2482, 2549, etc}.