POINTS ON CUBICS

This continuation of Points on Cubics has two sections, accessible by scrolling down or clicking:

H(U,P): Hirst Cubics
HP(U,P): H-Plus Cubics

Access to related material is provided at the bottom of this page.

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H(U,P): Hirst Cubics

defined by

   [(rv²w - qw²u)y - (qw²v - rv²u)z]x²
+ [(pw²u - ru²v)z - (ru²w - pw²v)x]y²
      + [(qu²v - pv²w)x - (pv²u - qu²w )y]z² = 0

Locus: The cubic H(U,P) is the locus of a point X = x : y : z such that the U-Hirst inverse of X is on the line PQ, where Q is the trilinear product U*X.

Notes:

1.   H(U,P) is also given by

[(pvw - ru²)wy² + (pvw - qu²)vz²]x + [(qwu - pv²)uz² + (qwu - rv²)wx²]y + [(ruv - qw²)vx² + (ruv - pw²)uy²]z = 0.

2.   The U-Hirst inverse of X is the point given by

Φ(U,X) = vwx² - yzu² : wuy² - zxv² : uvz² - xyw²,

so that

(U-Hirst inverse of X) = (X-Hirst inverse of U),

a commutativity comparable to that of the U-isoconjugate of X, defined by

vwyz : wuzx : uvxy.

X(1)-Hirst inversion is likened to X(1)-isoconjugation (i.e., isogonal conjugation). Their interaction regarding cubics is suggested by Notes 3 and 9, as well as the following identity:

H(X(1),P) = ZP(P ^-1, Q - R),

where denotes the crossdifference of X(1) and P.

3.   Four points on H(U,P) are the following: U, the P ^-1-isoconjugate of U, the P-Hirst inverse of U, and the point

[Φ(X(1),U ^-1)] ^-1 = u/(u² - vw) : v/(v² - wu) : w/(w² - uv).

4.   H(U,P) meets the sidelines BC, CA, AB in these points:

0 : pv² - qwu : pw² - ruv,    qu² - pvw : 0 : qw² - ruv,    ru² - pvw : rv² - qwu : 0.

5.   If X is on H(U,P), then X* is on H(U*,P), where X* and U* are the P ^-1-isoconjugates of X and U.

6.   H(U,P) is degenerate as the union of a line and a conic for some choices of (U,P); for example, H(X(1),P) is the union of the line

(q - r)α + (r - p)β + (p - q)γ = 0

and the Steiner circumellipse. H(U,P) is degenerate if P = U and also for these choices of (U,P):

(X(238),X(1)), (X(239),X(1)), (X(6),X(101)), (X(55),X(101)), (X(100),X(101)), (X(651),X(101)).

7.   If P = U, the defining equation yields

(x/u + y/v + z/w)[(1/v - 1/w)/x + (1/w - 1/u)/y + (1/u - 1/v)/z] = 0,

showing that H(U,U) is degenerate.

8.   The trilinear product U*U is on H(U,X(1)).

9.   H(U,X(1)) = Z(Φ(U,X(1))), U ^-1), where Φ is as in Note 2. Examples:

H(X(6),X(1)) = Z(X(238),X(2)),   H(X(2),X(1)) = Z(X(239),X(6)).

10.  If the defining equation is written as Ψ(U,P,X) = 0, then the same cubic is given by Ψ(X,P,U) = 0. That is, the defining equation is symmetric in U and X.

11.  In the definition of H(U,P), if the line P-to-U*X is replaced by P-to-X, then the resulting collinearity-determinant factors as a multiple of

(u/x + v/y + w/z)[(vr - wq)x + (wp - ur)y + (uq - vp)z],

so that the locus of such a point X is a degenerate cubic.

12.  Problem: For given U, determine those P for which the three terms in the defining equation for H(U,P) are each identically zero. It is easy to check that if P = X(1), then for each U, the unique solution is the aforementioned point
[Φ(X(1),U ^-1)] ^-1.

For U = X(2), choices of P with solution X are given by P = X(I) and X = X(J) for these (I,J):
(1, 291), (8,518), (10,1), (76,239), (85,673), (257,6), (274,2), (330,1575).

For U = X(6), choices of P with solution X are given by P = X(I) and X = X(J) for these (I,J):
(1,292), (32,238), (41,105), (56,672), (58,6), (172,31), (213,1), (904),2).

13.  The discriminant d of H(U,P), obtained by solving the defining equation for x in terms of y and z, is given by

d² = [wy²(pvw - ru²) - vz²(pvw - qu²)]² + 4uyzF₁F₂,  where

F₁ = (quw² - rwv²)y - (ruv² - qvw²)z,    and    F₂ = (pw² - ruv)y - (pv² - qwu)z.

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H(X(2),X(1)) passes through these centers:
1, 2, 6, 75, 239, 291, 366, 518, 673, 1575, 2319, 2p1423, 2p1932, 2p2054, 2p2106, 2p2107, 2p2108, 2p2109, 2p2110, 2p2111, 2p2112, 2p2114, 2p2115, 2p2116, 2p2117, 2p1118, 2p1119, 2p2144, 2p2145, 2p2146, 2p2147, 75p727, 694p1281, 1655p2248

H(X(2),X(1)) = Z(X(239),X(6)). See the latter for collinear triples on this cubic.

---

H(X(2),X(10)) passes through these centers:
1, 2, 7, 37, 291, 335, 350, 518, 694, 1909, 2p1757, 8p292, 10p2248, 28p337, 87p1575, 105p335, 239p2113, 291p292, 291p1929, 292p1281, 726p2109

Collinear triples

1	2	2p1757
1	37	518
1	291	8p292
1	350	1909
2	7	292p1281
2	37	350
2	291	335
2	1909	87p1575
7	350	8p292
7	518	1909
37	350	28p337
37	694	8p292
37	10p2248	292p1281
291	518	291p292
291	1909	28p337
291	291p1929	292p1281
335	350	291p292
335	518	105p335
335	694	1909
335	2p1757	291p292
350	2p1757	726p2109
350	8p292	105p1281
350	239p2113	292p1281
518	2p1757	239p2113
694	291p292	292p1281
1909	2p1757	10p2248
2p1757	28p337	291p292
8p292	87p1575	291p292
28p337	105p335	292p1281
87p1575	292p1281	726p2109

---

H(X(2),X(63)) passes through these centers:
2, 3, 57, 291, 447, 517, 1944

Collinear triples

2	3	447
2	57	1944
3	57	517

---

H(X(2),X(76)) passes through these centers:
2, 8, 75, 239, 256, 274, 291, 350, 740, 1281, 2p673, 2p1447, 2p1929, 2p1966, 86p1654, 238p1921, 239p518, 334p2112, 350p1757

Collinear triples

2	8	239
2	75	350
2	274	2p1966
2	291	239p518
2	1281	2p1929
8	75	2p1447
8	256	740
8	2p673	238p1921
75	256	2p1966
75	274	740
75	291	238p1921
75	2p673	239p518
75	2p1929	350p1757
239	274	86p1654
239	291	334p2112
239	350	350p1757
239	740	239p518
239	2p1966	238p1921
256	291	1281
256	350	86p1654
274	291	350p1757
274	1281	2p673
350	740	238p1921
350	1281	2p1447
350	2p673	334p2112
740	2p1929	334p2112
2p1447	2p1966	239p518
2p1929	86p1654	238p1921

---

H(X(2),X(85)) passes through these centers:
2, 7, 8, 279, 291, 508, 516, 673, 2p241, 2p1279, 2p1447, 2p2114, 7p1280, 85p103, 85p2115, 330p1742

Collinear triples

2	7	2p241
2	8	2p1279
7	8	2p1447
7	279	516
7	673	2p2114
8	279	2p241
8	2p2114	85p103
279	291	2p2114
279	2p1447	330p1742
291	279	2p1447
516	673	2p1279
516	2p241	85p103
2p241	2p1279	7p1280
2p241	2p2114	85p2115

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H(X(2),X(92)) passes through these centers:
2, 4, 278, 291, 447, 515, 1948

Collinear triples

2	4	447
2	278	1948
4	278	515

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H(X(3),X(1)) passes through these centers:
1, 3, 4, 255, 296, 416, 1936, 3p415, 90p1745, 1046p1248

Collinear triples

1	3	1936
3	4	416
4	225	1936
255	416	1046p1248

---

H(X(3),X(101)) passes through these centers:
3, 109, 296, 651, 1783, 2196, 2338, 3p1020

Collinear triples

3	296	3p1020
651	1783	3p1020

---

H(X(4),X(1)) passes through these centers:
1, 3, 4, 158, 243, 415, 1937, 4p416, 90p1148, 1047p1247

Collinear triples

1	4	243
1	158	4p416
3	4	415
3	158	243
158	415	1047p1247
415	1937	4p416

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H(X(6),X(1)) passes through these centers:
1, 2, 6, 31, 105, 238, 292, 365, 672, 1423, 1931, 2053, 2054, 2106, 2107, 2108, 2109, 2110, 2111, 2112, 2113, 2114, 2115, 2116, 2117, 2118, 2119, 2144, 2145, 2146, 2147, 2p727, 6p1575, 1045p2248, 1281p1967

H(X(6),X(1)) = Z(X(238),X(2)). See the latter for a table of collinear triples on this cubic.

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H(X(6),X(3)) passes through these centers:
6, 55, 58, 63, 103, 292, 511, 672, 1423, 3p862, 8p904

Collinear triples

6	55	672
6	58	511
55	3p862	8p904
58	63	3p862
63	672	1423
511	1423	8p904

---

H(X(6),X(32)) passes through these centers:
6, 31, 58, 238, 292, 727, 893, 1691, 1914, 2176, 2195, 3p862, 6p2068, 6p2069, 58p1045, 81p2107, 87p1914, 105p2110, 238p1458, 238p1914, 726p2210

Collinear triples

6	31	1914
6	58	1691
6	238	2176
6	292	726p2210
6	2195	238p1914
31	58	238p1914
31	292	740
31	727	726p2210
31	893	1691
31	2176	87p1914
31	2195	238p1458
31	6p2068	6p2069
58	238	58p1045
238	1691	238p1914
238	2195	105p2110
238	3p862	726p2210
292	1914	105p2110
292	2176	238p1458
727	2176	238p1914
893	1914	58p1045
893	2176	3p862
1691	1914	238p1458
1691	81p2107	105p2110
1691	87p1914	726p2210
1914	3p862	238p1914
58p1045	81p2107	238p1914

---

H(X(6),X(36)) passes through these centers:
6, 292, 840, 902, 1423, 2163, 2p36

Collinear triples

6	672	902
672	1423	2p36
902	2163	2p36

---

H(X(6),X(40)) passes through these centers:
6, 292, 672, 937, 972, 1423, 2p40, 2p1911, 3p55

Collinear triples

6	212	672
329	672	1423

---

H(X(6),X(41)) passes through these centers:
6, 55, 56, 105, 292, 1914, 2110, 2223, 2p1911, 6p2111

Collinear triples

6	55	1914
6	292	2110
55	56	2223
55	105	2110
292	1914	2223
2110	2223	6p2111

---

H(X(6),X(48)) passes through these centers:
3, 6, 56, 104, 292, 1951, 2223, 2p1911, 48p422, 956p2279

Collinear triples

3	6	48p422
3	56	2223
6	56	1951
6	2223	956p2279

---

H(X(6),X(56)) passes through these centers:
6, 56, 57, 292, 672, 1423, 1458, 1477, 2195, 2223, 2p1911, 6p1743, 57p1362, 87p2223, 103p1419, 103p2114, 238p1458, 672p1279

Collinear triples

6	56	1458
6	672	6p1743
6	2195	672p1279
6	57p1362	103p1419
56	57	2223
56	292	238p1458
56	1324	87p2223
56	1477	672p1279
56	2195	57p1362
57	672	1423
57	2195	238p1458
292	1423	57p1362
672	1458	238p1458
672	2223	672p1279
672	103p1419	103p2114
238	2195	103p2114
1458	2195	103p2114
1458	2223	57p1362
1477	6p1743	57p1362

---

H(X(6),X(58)) passes through these centers:
6, 58, 81, 292, 741, 1326, 1911, 2106, 2p1911, 3p862, 292p1929, 741p2107

Collinear triples

6	58	1326
6	81	2106
6	292	1911
58	81	3p862
58	292	741
81	741	1911
292	3p862	292p1929
741	1326	292p1929
741	2106	3p862
1911	3p862	741p2107

---

H(X(7),X(1)) passes through these centers:
7, 55, 174, 241, 673, 1088, 2p1742, 503p1488

Collinear triples

7	55	174
174	241	503p1488
241	1088	2p1742

---

H(X(7),X(85)) passes through these centers:
7, 8, 279, 291, 508, 516, 673, 2p241, 2p1279, 2p1447, 7p1280, 85p103, 85p2115, 330p1742

Collinear triples

2	7	2p241
2	8	2p1279
7	8	2p1447
7	279	516
7	673	2p2114
8	279	2p241
8	2p2114	85p103
279	291	2p2114
279	2p1447	330p1742
291	2p241	2p1447
516	673	2p1279
516	2p241	85p103
2p241	2p1279	7p1280
2p241	2p2114	85p2115

---

H(X(9),X(1)) ( = Z(X(518),X(57)) ) passes through these centers:
1, 9, 57, 200, 239, 294, 518, 1280, 1282, 2348, 2p103, 2p2115, 8p1911, 9p509, 8p910, 1742p2319

Collinear triples

1	9	518
1	57	9p910
1	200	239
1	294	1289
9	200	2348
57	200	5181
57	239	9p509
57	1282	8p1911
200	1282	2p103
239	518	8p1911
294	2348	9p910
518	1280	2348
518	1282	2p2115
518	2p103	9p910

---

H(X(19),X(1)) passes through these centers:
1, 19, 63, 240, 1096, 1910, 1967, 2312, 2p419, 2p1692, 4p401, 4p1297, 25p147, 64p193, 297p2065

Collinear triples

1	19	240
1	63	2p1692
1	1096	4p401
19	63	2p419
19	1096	2312
19	1910	25p147
63	240	1096
63	2312	64p193
63	4p1297	25p147
240	1967	2p419
240	2312	4p1297
240	2p1692	297p2065
240	4p401	6p1987
1096	1967	25p147
1910	2312	2p1692
1910	2p419	4p401

---

H(X(31),X(1)) passes through these centers:
1, 6, 31, 75, 560, 1403, 1580, 1755, 1910, 1967, 2053, 2p699, 4p1613, 25p147, 31p2227, 83p2076, 1281p1911

Collinear triples

1	31	1580
1	75	31p2227
1	560	83p2076
1	1403	1281p1911
6	560	1281p1911
6	1403	1755
6	2053	31p2227
31	560	1755
75	560	1580
75	1755	4p1613
560	1910	25p147
1403	1580	2053
1580	1755	1910
1580	2p699	31p2227
1755	1967	31p2227
1967	4p1613	25p147
2053	4p1613	1281p1911

---

H(X(31),X(32)) passes through these centers:
6, 31, 213, 1911, 1922, 1967, 2210, 6p172, 6p741, 6p1458, 6p2162, 31p1575, 43p1911, 213p2106, 291p1911, 292p727, 292p2195, 741p2107, 1911p2068, 1911p2069, 2111p2223

Collinear triples

6	31	213p2106
6	213	31p1575
6	1911	43p1911
6	2210	6p172
31	213	2210
31	1911	1922
31	6p172	6p1458
213	1922	6p741
213	1967	43p1911
1911	2210	292p2195
1911	6p172	6p741
1911	31p1575	291p1911
1922	1967	6p172
1922	2210	291p1911
1922	6p1458	292p2195
1922	6p2162	43p1911
1922	31p1575	292p727
1922	213p2106	741p2107
1922	1911p2068	1911p2069
1967	213p2106	292p2195
2210	43p1911	292p727
2210	213p2106	2111p2223
6p172	6p2162	31p1575
6p741	213p2106	291p1911
6p1458	43p1911	291p1911

---

H(X(37),X(1)) passes through these centers:
1, 37, 81, 291, 756, 1654, 1757, 2238, 2p2054, 2p2108, 37p2109, 37p2248

Collinear triples

1	37	1757
37	756	2238
81	756	1757
82	1654	2238
291	756	2p2108
291	1757	2238
1654	1757	37p2248
1654	2p2054	2p2108
1757	2p2108	37p2109

---

H(X(37),X(10)) passes through these centers:
2, 10, 37, 210, 740, 1654, 2238, 2p1284, 2p2054, 10p105, 10p385, 10p904, 10p1911, 37p2238, 42p1281, 239p740, 518p2238, 740p1757, 1655p2107

Collinear triples

2	10	37p2238
2	37	10p385
2	1654	22381
2	10p105	42p1281
2	10p1911	740p1757
2	239p740	1655p2107
10	37	740
10	210	2p1284
10	2p2054	740p1757
10	10p1050	518p2238
10	10p385	10p904
10	10p1911	239p740
37	210	2238
37	2p2054	42p1281
37	10p1911	518p2238
210	10p105	239p740
210	10p904	37p2238
740	1654	10p904
740	2238	740p1757
740	2p1284	42p1281
740	37p2238	239p740
1654	2p2054	239p740
2238	10p385	239p740
2238	37p2238	518p2238
2p1284	10p385	518p2238
10p904	10p1911	42p1281
10p904	518p2238	1655p2107

---

H(X(42),X(1)) passes through these centers:
1, 42, 86, 292, 872, 2107, 2110, 2238, 6p1045, 37p2106, 37p2111

Collinear triples

1	42	37p2106
42	872	2238
86	872	37p2106
86	2238	6p1045
292	872	2110
292	2238	37p2106
2107	2110	6p1045
2110	37p2106	37p2111

---

H(X(42),X(9)) passes through these centers:
42, 333, 756, 2107, 2161, 2238, 2258, 6p1045

Collinear triples

42	756	2238
333	2238	6p1045

---

H(X(55),X(1)) passes through these centers:
1, 7, 55, 259, 294, 672, 1253, 9p1742

Collinear triples

7	672	9p1742
57	672	1243

---

H(X(57),X(1)) passes through these centers:
1, 9, 57, 105, 241, 269, 292, 509, 910, 1279, 1447, 2114, 2p1477, 7p103, 7p2115, 87p,1742, 355p1355

Collinear triples

1	9	1279
1	57	241
9	57	1447
9	241	269
9	2114	7p103
57	105	2114
57	269	910
105	910	1279
241	292	1447
241	910	7p103
241	1279	2p1477
241	2114	7p2115
269	292	2114
269	1447	87p1742

---

H(X(57),X(56)) passes through these centers:
1,6, 57, 105, 1279, 1429, 1462, 2162, 2p1462, 7p1458, 105p291, 105p294, 105p1743, 192p1462, 269p1376, 673p1477

Collinear triples

1	6	1279
1	57	1429
1	105	105p1743
1	2p1462	192p1462
1	7p1458	269p1376
6	57	7p1458
6	1462	2p1462
57	105	1462
105	1279	105p294
105	2p1462	269p1376
1279	1462	673p1477
1429	1462	105p291
1429	2162	269p1376
1429	2p1462	105p294
1462	2162	192p1462
1462	7p1458	105p294
7p1458	105p291	192p1462
7p1458	105p1743	673p1477

---

H(X(63),X(1)) passes through these centers:
1, 19, 63, 293, 326, 1959, 2p401, 2p1297, 3p230, 3p694, 3p1503, 6p147, 69p419, 193p1073, 194p1988, 325p2065

Collinear triples

1	19	3p1503
1	63	1959
1	293	6p107
19	63	2p401
19	326	1959
19	3p694	6p147
63	326	3p230
293	3p230	3p1503
326	2p401	194p1988
326	2p1297	6p147
326	3p1503	193p1073
1959	2p401	3p1987
1959	2p1297	3p1503
1959	3p230	325p2065

---

H(X(65),X(1)) passes through these centers:
1, 21, 65, 851, 1254, 1758, 1937, 4p1046, 73p1247

Collinear triples

1	65	1758
21	851	4p1046
21	1254	1758
65	851	1254
851	1758	1937
1758	4p1046	73p1247

---

H(X(75),X(1)) passes through these centers:
1, 2, 31, 75, 561, 1581, 1821, 1959, 1966, 2227, 2p1423, 2p2319, 4p147, 4p194, 76p699, 291p1281, 308p2076

Collinear triples

1	31	1959
2	31	291p1281
2	1959	2p1423
2	2227	2p2319
31	75	308p2076
31	1821	4p147
75	561	2227
75	2p1423	291p1281
561	1959	4p194
1581	1959	2227
1581	4p147	4p194
2p2319	4p147	291p1281

---

H(X(75),X(76)) passes through these centers:
2, 10, 75, 85, 334, 335, 1581, 1920, 1921, 2p518, 8p291, 27p337, 75p1757, 87p726, 105p334, 291p291, 291p1281, 321p2248, 335p1929, 350p2113

Collinear triples

2	10	2p518
2	75	75p1753
2	335	8p291
2	1920	1921
10	75	1921
10	334	27p337
10	1581	8p291
10	291p1281	321p2248
75	85	291p1281
75	334	335
75	1920	87p726
85	334	8p291
85	1920	2p518
334	1581	1920
334	1921	291p291
334	2p518	105p334
334	75p1757	335p1929
335	1920	27p337
335	2p518	291p291
335	291p1281	335p1929
1581	291p291	291p1281
1920	75p1757	321p2248
1921	8p291	105p334
1921	291p1281	350p2113
2p518	75p1757	350p2113
8p291	87p726	291p291
27p337	75p1757	291p291
27p337	105p334	291p1281

---

H(X(81),X(1)) passes through these centers:
1, 37, 81, 238, 757, 1929, 1931, 2109, 2248, 2p741, 81p1654, 86p2108

Collinear triples

1	37	238
1	81	1931
37	757	1931
37	2p741	86p2108
238	757	81p1654
238	1931	2p741
1929	81p1654	86p2108
1931	2109	86p2108
1931	2248	81p1654

---

H(X(105),X(1)) passes through these centers:
1, 6, 57, 105, 294, 518, 673, 910, 1376, 87p238, 103p673, 105p105, 105p165, 673p1742, 1447p2115

Collinear triples

1	6	518
1	105	294
6	57	910
6	294	673
6	1376	87p238
57	105	105p165
57	518	1376
57	673	673p1742
105	673	1376
105	910	105p105
294	518	105p105
294	910	103p673
518	910	1447p2115
518	103p673	105p165
87p238	105p105	673p1742

---

H(X(240),X(1)) passes through these centers:
1, 19, 92, 240, 293, 1581, 1707, 1755, 1959, 4p194, 6p114, 132p1073, 232p385, 232p297, 282p385, 240p1957, 262p1351, 297p2065, 1249p1297

Collinear triples

1	19	240
1	92	240p1957
1	293	6p114
1	1707	1959
1	1755	262p1351
1	232p297	1249p1297
19	92	1755
19	293	232p297
19	1581	232p385
19	6p114	297p2065
19	132p1073	1249p1297
92	293	232p285
92	1959	4p194
240	1755	232p297
240	1959	232p385
240	132p1073	240p1957
293	1707	132p1073
1581	4p194	232p297
1707	232p297	297p2065
1755	1959	6p114
1959	232p297	240p1957
232p385	262p1351	297p2065

---

H(X(241),X(1)) passes through these centers:
1, 7, 57, 241, 291, 294, 518, 672, 1743, 2p1423, 2p1477, 7p1362, 87p672, 238p241, 241p2348

Collinear triples

1	57	241
1	294	241p2348
1	518	1743
7	57	672
7	294	238p241
7	518	2p1423
57	291	238p241
57	294	7p1362
57	2p1423	87p672
57	2p1477	241p2348
241	518	238p241
241	672	7p1362
291	2p1423	7p1362
518	672	241p2348
1743	2p1477	7p1362

---

H(X(243),X(1)) passes through these centers:
1, 4, 29, 46, 243, 296, 851, 1936, 4p450, 29p1046, 90p243, 243p243, 243p1758, 851p1247, 1936p1942

Collinear triples

1	4	243
1	29	4p450
1	46	1936
4	29	851
4	46	90p243
4	296	243p243
4	4p450	1936p1942
4	29p1046	851p1247
29	296	243p1758
29	1936	29p1046
46	851	1936p1942
243	851	243p243
243	1936	243p1758
243	29p1046	1936p1942
1936	4p450	243p243
4p450	243p1758	851p1247

---

H(X(293),X(1)) passes through these centers:
1, 48, 63, 240, 293, 1580, 1821, 1910, 1957, 2p2065, 3p230, 3p230, 64p441, 98p194, 193p248, 248p1916, 401p1988

Collinear triples

1	48	240
1	63	1580
1	293	1910
1	1957	64p441
48	63	3p230
48	1821	1910
48	1957	401p1988
63	240	1957
63	293	193p248
63	1821	98p194
240	1910	248p287
240	2p2065	193p248
240	98p194	248p1916
293	1821	1957
293	3p230	248p287
1580	1821	248p287
1580	1910	248p1916
1910	2p2065	3p230
64p441	193p248	248p287
98p194	248p287	401p1988

---

H(X(294),X(1)) passes through these centers:
1, 9, 55, 105, 238, 241, 294, 673, 2053, 2348, 43p673, 57p1376, 105p1280, 145p2195, 291p294, 294p294, 514p1293, 516,2115

Collinear triples

1	9	238
1	55	241
1	105	294
9	55	2348
9	241	57p1376
9	294	145p2195
9	673	43p673
55	105	673
105	238	291p294
105	241	294p294
105	2053	43p673
105	2348	105p1280
105	145p2145	514p1293
238	241	516p2115
238	673	294p294
238	2053	57p1376
241	43p673	291p294
241	105p1280	145p2195
294	673	57p1376
294	2348	294p294
2348	57p1376	514p1293

---

H(X(296),X(1)) passes through these centers:
1, 3, 73, 90, 243, 296, 1758, 1937, 1940, 2p2249, 3p1942, 46p296, 73p1247, 296p292

Collinear triples

1	3	1758
1	73	243
1	296	1937
3	243	1940
3	296	46p296
73	1937	2p2249
73	3p1942	46p296
90	1937	46p296
243	1937	296p296
296	1940	2p2249
1758	1940	73p1247
1758	2p2249	296p296
1937	1940	3p1942

---

H(X(518),X(1)) passes through these centers:
1, 2, 9, 105, 165, 241, 518, 672, 2p103, 2p1742, 8p1362, 43p291, 59p1566, 291p1282, 518p1376, 1280p1743

Collinear triples

1	2	518p1376
1	9	518
1	105	59p1566
1	165	241
1	8p1362	1280p1743
2	9	672
2	241	2p1742
2	8p1362	43p291
9	105	8p1362
9	2p103	59p1566
105	672	291p1282
165	2p103	8p1362
241	672	59p1566
241	8p1362	518p1376
241	43p291	291p1282
518	672	8p1362
2p103	291p1282	518p1376

---

H(X(672),X(1)) passes through these centers:
1, 6, 55, 241, 292, 518, 672, 673, 55p1362, 238p518, 291p2110

Collinear triples

1	6	518
1	55	241
1	292	238p518
1	673	55p1362
6	55	672
6	292	55p1362
55	673	238p518
241	518	55p1362
292	672	291p2110
518	672	238p518
518	673	291p2110

---

H(X(673),X(1)) passes through these centers:
1, 2, 7, 105, 239, 294, 672, 673, 105p291, 239p2111, 673p673

Collinear triples

1	2	239
1	105	294
2	7	672
2	105	673
7	294	673
105	672	105p291
239	294	673p673
239	672	239p2111
239	673	105p291
672	673	673p673

---

---

HP(U,P): H-Plus Cubics,

defined by

   [(rv²w + qw²u)y - (qw²v + rv²u)z]x²
+ [(pw²u + ru²v)z - (ru²w + pw²v)x]y²
      + [(qu²v + pv²w)x - (pv²u + qu²w )y]z² = 0

The Hirstpoint of U and X is here introduced by trilinears

vwx² + yzu² : wuy² + zxv² : uvz² + xyw².

Hirstpoint, cevapoint, and crosspoint are commutative operations associated with Hirst inverse, Ceva conjugate, and cross conjugate. The Hirstpoint of U and X is the harmonic conjugate of the U-Hirst inverse of X with respect to the points

vwx² : wuy² : uvz²    and    yzu² : zxv² : xyw².

Locus:   The cubic HP(U,P) is the locus of a point X = x : y : z such that the Hirstpoint of U and X lies on the line PQ, where Q denotes the trilinear product U*X.

Notes:

1.   HP(U,P) is also given by

[(pvw + ru²)wy² - (pvw + qu²)vz²]x + [(qwu + pv²)uz² - (qwu + rv²)wx²]y + [(ruv + qw²)vx² - (ruv + pw²)uy²]z = 0.

2.   HP(X(1),P) = Z((Λ(P)) ^-1, Λ(P)), where Λ(P) denotes the crosssum of X(1) and P.
For example, HP(X(1),X(2)) = Z(X(86), X(42)).

3.   HP(U,P) meets the sidelines BC, CA, AB in these points:

0 : pv² + qwu : pw² + ruv,    qu² + pvw : 0 : qw² + ruv,    ru² + pvw : rv² + qwu : 0

4.   If the defining equation is written as Ψ(U,P,X) = 0, then the same cubic is given by Ψ(X,P,U) = 0. That is, the defining equation is symmetric in U and X.

5.   The cubic HP(U,U) is given by t₁ + t₂ + t₃ = 0, where

t₁ = vw[w(u + v)y - v(u + w)z)x²,

and t₂ and t₃ are defined cyclically from t₁. The point on HP(U,U) for which t₁ = 0 (and t₂ = t₃ = 0) is the crosspoint of X(1) and U.

---

HP(X(1),X(2)) ( = Z(X(86),X(42)) ) passes through these centers:
1, 2, 7, 21, 29, 77, 81, 86, 2p1433, 4p1817, 20p84, 253p2360, 1034p1394

For collinear triples, see Z(X(86),X(42)).

---

HP(X(1),X(20)) passes through these centers:
1, 8, 20, 21, 29, 78, 280, 1043, 3p1034, 4p1819, 29p1490, 64p1043, 271p1712,

Collinear triples

1	8	78
1	21	4p1819
1	29	29p1490
1	280	271p1712
8	20	280
8	21	1043
8	29	4p1819
8	3p1034	271p1712
8	29p1490	64p1043
20	21	29
20	78	3p1034
20	1043	64p1043
21	280	29p1490
21	64p1043	271p1712
29	78	1043
78	4p1819	64p1043
280	1043	4p1819
1043	3p1034	29p1490

---

HP(X(1),X(40)) ( = Z(X(8),X(56)) ) passes through these centers:
1, 8, 40, 175, 176, 188, 280, 483, 2p2066, 8p2067, 8p2362, 9p557, 92p2066, 164p188, 188p505
For collinear triples, see Z(X(8),X(56))

---

HP(X(1),X(57)) ( = HP(X(1),X(40)) ) passes through these centers:
1, 2, 7, 57, 145, 174, 1488, 2089, 145p2137

Collinear triples

1	2	145
1	174	2089
2	7	57
7	1488	2089
57	145	145p2137
57	174	1488

---

HP(X(1),X(63)) ( = Z(X(75),X(31)) ) passes through these centers:
1, 2, 7, 8, 63, 75, 92, 280, 347, 1895, 2p1073, 189p1490, 223p1034, 253p1498, 1032p1249
For collinear triples, see Z(X(75),X(31))

---

HP(X(2),X(1)) ( = Z(X(894),X(6)) ) passes through these centers:
6, 7, 9, 37, 75, 86, 87, 192, 256, 366, 894, 1045, 1654, 2p2248, 1575p1929
For collinear triples, see Z(X(894),X(6))

---

HP(X(2),X(2)) ( = Z(X(2),X(58)) ) passes through these centers:
1, 2, 9, 10, 37, 226, 281, 1214, 2p2331, 10p1433, 20p1903, 40p2184, 1490p2184
For collinear triples, see Z(X(2),X(58))

---

HP(X(3),X(1)) passes through these centers:
4, 21, 73, 90, 255, 1046, 1248, 1745, 1935, 3p46, 3p1047, 3p1247, 1936p1942

Collinear triples

4	21	3p1047
4	73	1745
4	255	1935
21	73	1935
21	255	1046
73	255	3p46
90	1935	3p46
1046	1745	1936p1942
1046	1935	3p1247
1248	1935	3p1047
3p46	3p1047	136p1942

---

HP(X(3),X(3)) passes through these centers:
1, 3, 73, 1745, 3p73

Collinear triples

1	3	3p73
1	73	1745

---

HP(X(4),X(1)) passes through these centers:
3, 29, 46, 65, 158, 1047, 1148, 1247, 1940, 4p90, 4p1046, 4p1248, 243p1942

Collinear triples

3	29	4p1046
3	46	65
3	158	1940
29	65	1940
29	158	1047
46	1047	243p1942
46	1940	4p90
65	159	1148
1047	1940	4p1248
1148	4p1046	243p1942
1247	1940	4p1046

---

HP(X(4),X(4)) passes through these centers:
1, 4, 46, 65, 225, 2362, 10p2362, 46p254, 90p225, 225p1806, 485p2362

Collinear triples

1	4	225
1	46	65
4	46	90p225
4	10p2362	225p1806
46	225	46p254
46	10p2362	485p2362
65	2362	225p1806
225	2362	485p2362

---

HP(X(5),X(5)) passes through these centers:
1, 5, 5p35, 5p65, 5p73, 5p2307, 319p2181, 484p1263, 559p1953, 655p2081

Collinear triples

5	5p35	5p65
5	5p73	319p2181
5p65	5p73	655p2081
5p73	5p2307	559p1953

---

HP(X(6),X(1)) ( = Z(X(171),X(2)) ) passes through these centers:
2, 31, 42, 43, 55, 57, 81, 171, 365, 846, 893, 2162, 2248, 6p1045, 1282p2111

Collinear triples

2	31	171
2	42	43
2	81	6p1045
31	42	55
31	81	846
42	81	171
43	171	2162
43	846	893
55	57	171
55	893	6p1045
171	846	2248

---

HP(X(6),X(6)) passes through these centers:
1, 6, 42, 43, 213, 42p2162

Collinear triples

1	6	213
1	42	43
6	43	42p2162

---

HP(X(7),X(7)) passes through these centers:
1, 7, 57, 142, 354, 2p1418, 7p2191, 142p218

Collinear triples

1	7	2p1418
1	57	354
7	57	142
7	7p2191	142p218
57	2p1418	7p2191
142	354	142p218

---

HP(X(8),X(1)) passes through these centers:
40, 56, 188, 280, 341, 979, 1050, 1222, 2p2347, 8p978

Collinear triples

40	56	2p2347
341	1050	1222
341	2p2347	8p978

---

HP(X(8),X(8)) passes through these centers:
1, 8, 40, 2p2347, 40p2123, 312p2347

Collinear triples

1	8	312p2347
1	40	2p2347
40	40p2123	312p2347

---

HP(X(9),X(1)) passes through these centers:
2, 43, 55, 57, 165, 200, 1376, 2319, 9p509, 9p1742, 9p1743, 103p239, 514p1293

Collinear triples

2	43	200
2	55	1376
2	57	9p1742
43	1376	2319
55	57	165
55	200	9p1743
57	200	1376
165	200	103p239
1376	9p1743	514p1293

---

HP(X(9),X(9)) passes through these centers:
1, 9, 55, 165, 220, 2066, 9p365, 9p2067, 55p1489, 165p2125, 200p2362, 281p2066

Collinear triples

1	9	220
1	55	165
9	2066	200p2362
9	9p2067	281p2066
55	2066	9p2067
165	220	165p2125
220	200p2362	281p2066

---

HP(X(10),X(10)) passes through these centers:
1, 10, 191, 2292, 10p2067, 10p2292, 12p1806, 191p2127, 502p2292

Collinear triples

1	10	10p2292
1	191	2292
10	191	502p2292
191	10p2292	191p2127
2292	10p2067	12p1806

---

HP(X(13),X(13)) passes through these centers:
1, 13, 3p1832, 10p2306, 13p202, 45p1987, 196p249, 967p1972, 1081p1956, 1807p1833, 1956p1987

Collinear triples

1	13	10p2306
1	3p1832	13p202
13	13p202	1807p1833
45p1987	967p1972	1081p1956
45p1987	967p1972	1956p1987
45p1987	1081p1956	1956p1987
196p249	1081p1956	1956p1987
967p1972	1081p1956	1956p1987

---

HP(X(14),X(14)) passes through these centers:
1, 14, 3p1833, 14p203, 37p554, 1807p1832

Collinear triples

1	14	37p554
1	3p1833	14p203
14	14p203	1807p1832

---

HP(X(19),X(1)) passes through these centers:
31, 63, 92, 204, 1096, 1707, 1957, 1957, 2184, 4p1613, 4p1988, 230p1987, 419p1297, 694p1503

Collinear triples

31	63	1707
31	92	1957
31	204	1096
63	92	4p1613
63	204	419p1297
63	1096	1957
204	1057	2184
1096	4p1613	694p1503
1957	419p1297	694p1503

---

HP(X(19),X(19)) passes through these centers:
1, 19, 31, 1707, 1973, 6p371, 6p372, 6p2362, 19p2066, 25p584, 25p486, 33p2067, 34p2066, 1707p2129

Collinear triples

1	19	1973
1	31	1707
19	6p371	25p485
19	6p372	25p486
19	6p2362	19p2066
19	33p2067	34p2066
31	6p371	6p372
31	6p2362	34p2066
1707	1973	1707p2129
1973	19p2066	33p2067
1973	25p485	25p486
6p371	19p2066	34p2066
6p372	6p2362	33p2067

---

HP(X(20),X(20)) passes through these centers:
1, 1394, 4p1394, 10p1394, 20p73, 20p207, 20p221, 20p1042, 1042p1097

Collinear triples

1	10p1394	20p207
1	20p73	20p221
1394	4p1394	1042p1097
1394	20p221	20p1042
4p1394	10p1394	20p221
4p1394	20p207	20p1042
10p1394	20p73	20p1042

---

HP(X(21),X(21)) passes through these centers:
1, 3, 21, 283p407, 407p1098

Collinear triples

1	3	283p407
3	21	283p1098

---

HP(X(24),X(24)) passes through these centers:
1, 24, 24p65, 24p73, 24p1062, 24p1479

Collinear triples

24	24p65	24p1062
24	24p73	24p1479

---

HP(X(25),X(25)) passes through these centers:
1, 25, 1402, 1716, 4p1402, 6p2082, 19p2082, 42p614, 1851p1918

Collinear triples

1	25	1851p1918
1	1716	42p614
25	1402	19p2082
25	4p1402	6p2082
1402	4p1402	42p614
6p2082	19p2082	1851p1918

---

HP(X(27),X(27)) passes through these centers:
1, 27, 57, 58, 278, 579, 1474, 1838, 2260, 2p942, 21p2260, 27p2260, 272p1841, 284p1838, 580p942

Collinear triples

1	27	27p2260
1	57	580p942
1	278	1838
1	579	2260
1	1474	284p1838
27	57	284p1838
27	58	1838
27	278	21p2260
27	579	272p1841
27	1474	2p942
57	58	27p2260
57	278	2260
57	579	2p942
58	579	21p2260
58	1474	2260
278	1474	272p1841
1838	2260	284p1838
1838	2p942	27p2260
1838	272p1841	580p942
2260	21p2260	580p942
21p2260	27p2260	284p1838

---

HP(X(28),X(28)) passes through these centers:
1, 28, 34, 56, 1104, 1724, 21p1104, 21p1842, 25p272, 28p1104, 404p1842

Collinear triples

1	28	28p1104
1	34	404p1842
1	1104	1724
28	34	21p1104
28	56	21p1842
34	56	1104
34	25p272	28p1104
1104	21p1842	404p1842
1724	21p1842	25p272
21p1104	21p1842	28p1104

---

HP(X(30),X(30)) passes through these centers:
1, 30, 30p36, 30p65, 30p73, 30p1464, 30p1870, 758p1354

Collinear triples

1	30	758p1354
30	30p36	30p65
30	30p73	30p1870
30p36	30p1870	758p1354
30p65	30p73	30p1464

---

HP(X(31),X(1)) passes through these centers:
6, 19, 48, 75, 82, 560, 1582, 1740, 1964, 6p695, 32p1031

Collinear triples

19	48	1582
48	560	1964
75	560	1582
75	1740	1964
82	1582	1964

---

HP(X(31),X(31)) passes through these centers:
1, 31, 1740, 1923, 1964

Collinear triples

1	31	1923
1	1740	1964

---

HP(X(33),X(33)) passes through these centers:
1, 33, 1721, 55p1836

Collinear triple

1

1721

55p1836

---

HP(X(34),X(34)) passes through these centers:
1, 34, 1722, 56p1837, 1398p1837

Collinear triples

1	34	1398p1837
1	1722	56p1837

---

HP(X(37),X(37)) passes through these centers:
1, 37, 846, 1962, 37p1251, 37p1962, 554p1334, 559p1962, 846p2135, 1082p1962

Collinear triples

1	37	37p1962
1	846	1962
37	37p1251	1082p1962
37	559p1962	559p1962
846	37p1962	846p2135
1962	559p1962	1082p1962
37p1251	37p1962	554p1334

---

HP(X(40),X(40)) passes through these centers:
1, 40, 221, 40p221, 40p266, 40p2067, 40p2362, 175p221, 176p221, 196p2066, 198p557, 198p558, 223p2066

Collinear triples

1	40	40p221
40	40p2067	196p2066
40	40p2362	223p2066
40	198p559	198p558
221	40p2067	223p2066
221	175p221	176p221
40p221	40p2362	196p2066
40p266	40p2067	198p558
40p266	198p557	223p2066
40p2067	40p2362	176p221
125p221	196p2066	223p2066

---

HP(X(55),X(55)) passes through these centers:
1, 55, 1742, 2293, 41p1212

Collinear triples

1	55	41p1212
1	1742	2293

---

HP(X(56),X(1)) passes through these centers:
8, 84, 221, 266, 978, 1106, 1201, 1476, 56p979, 56p1050

Collinear triples

8	978	1201
8	1476	56p1050
221	1106	1201

---

HP(X(56),X(56)) passes through these centers:
1, 56, 978, 1201, 31p1122

Collinear triples

1	56	31p1122
1	978	1201

---

HP(X(56),X(57)) passes through these centers:
2, 28, 57, 603, 978, 1201, 1407

Collinear triples

2	978	1201
28	57	603
603	1201	1407

---

HP(X(57),X(1)) passes through these centers:
6, 7, 9, 87, 269, 509, 1419, 1423, 1742, 1743, 57p1376, 103p1447, 291p910

Collinear triples

6	7	57p1376
6	9	1743
6	269	1419
7	9	1423
7	269	1742
9	269	57p1376
9	1419	103p1447
87	1423	57p1376
269	1423	291p910
57p1376	103p1447	291p910

---

HP(X(57),X(57)) passes through these centers:
1, 6, 56, 57, 266, 289, 1743, 6p2089, 1743p2137

Collinear triples

1	6	1743
1	56	57
6	266	6p2089
56	266	289
56	1743	1743p2137
57	289	6p2089

---

HP(X(63),X(1)) passes through these centers:
19, 48, 75, 326, 610, 1740, 1958, 2p1073, 2p1988, 3p193, 385p1297, 441p694

Collinear triples

19	48	610
19	326	1958
19	1740	441p694
48	75	1958
48	326	3p193
75	326	1740
326	610	385p1297
610	1958	2p1073
1958	385p1297	441p694

---

HP(X(63),X(63)) passes through these centers:
1, 3, 48, 63, 219, 222, 255, 268, 610, 3p223, 3p1073, 154p1032, 271p1035, 1073p1498

Collinear triples

1	3	3p223
1	48	610
1	63	255
1	222	271p1035
3	48	219
3	222	255
3	268	610
3	3p1073	271p1035
48	222	3p223
48	268	271p1035
48	3p1073	1073p1498
63	219	222
63	268	3p223
63	610	3p1073
63	154p1032	1073p1498
219	255	268
219	3p223	3p1073
219	154p1032	271p1035
222	268	1073p1498
255	610	154p1032

---

HP(X(65),X(65)) passes through these centers:
1, 65, 1046, 3p407, 73p407

Collinear triples

1	65	73p407
1	1046	3p407

---

HP(X(69),X(69)) passes through these centers:
1, 69, 72, 306, 1439, 1763, 2p1473, 7p1040, 8p1040, 10p1473, 20p1439, 307p1040, 1763p2139

Collinear triples

1	69	307p1040
1	306	8p1040
1	1763	10p1473
1	7p1040	20p14395
69	72	7p1040
69	306	2p1473
69	1439	8p1040
72	306	10p1473
72	1439	307p1040
1439	1763	2p1473
1439	10p1473	20p1439
1763	307p1040	1763p2139
2p1473	7p1040	307p1040
7p1040	8p1040	10p1473

---

HP(X(75),X(1)) passes through these centers:
2, 31, 38, 63, 92, 561, 1965, 2p83, 2p194, 2p695, 6p1031

Collinear triples

31	38	63
31	561	1965
38	561	2p194
38	1965	2p83
63	92	1965

---

HP(X(75),X(75)) passes through these centers:
1, 38, 63, 75, 1930, 2p427, 22p427, 66p69, 66p1370, 141p1342, 141p1343

Collinear triples

1	38	63
1	74	1930
1	2p427	66p1370
38	2p426	22p427
38	66p69	66p1370
38	141p1342	141p1343
63	75	2p427
63	1930	66p69
74	22p427	66p69

---

HP(X(81),X(1)) passes through these centers:
6, 37, 86, 757, 1100, 1963, 2p1171, 2p2248, 58p1654, 81p1051

Collinear triples

6	37	1100
6	86	1963
37	757	1963
37	2p1171	81p1051
757	1100	58p1654
1100	1963	2p1171
1963	2p2248	58p1654

---

HP(X(81),X(2)) passes through these centers:
1, 10, 58, 1029, 2p2363, 593p596, 1269p2220

Collinear triples

1	10	1269p2220
10	58	2p2363

---

HP(X(81),X(81)) passes through these centers:
1, 6, 81, 1100, 21p553, 58p1125, 596p1333, 1269p2220

Collinear triples

1	6	1100
1	81	58p1125
6	81	21p553
6	58p1125	596p1338
81	596p1333	1269p2220
1100	21p553	1269p2220

---

HP(X(91),X(91)) passes through these centers:
1, 91, 2p216, 3p486, 4p486, 4p637, 4p638, 5p68, 5p485, 5p486, 68p372

Collinear triples

1	91	5p68
91	3p486	5p485
91	5p486	68p372
2p216	3p486	68p372
2p216	4p637	4p638
3p486	4p637	5p486
4p638	5p485	68p372
5p68	5p485	5p486

---

HP(X(100),X(100)) passes through these centers:
1, 100, 513, 1381, 1382, 522p1381

Collinear triples

1	1381	1382
100	1382	522p1381

---

HP(X(101),X(101)) passes through these centers:
1, 41, 101, 663, 41p508, 513p1621

Collinear triples

1	41	101
1	663	513p1621

---

HP(X(105),X(105)) passes through these centers:
1, 105, 238, 1279, 105p1279

Collinear triples

1	105	105p1279
1	238	1279

---

HP(X(109),X(109)) passes through these centers:
1, 109, 603, 1459, 3p1381, 3p1382, 65p1822, 65p1823, 513p1381, 513p1382, 513p1822, 513p1823, 514p572

Collinear triples

1	109	603
1	1459	514p572
1	65p1822	65p1823
109	3p1381	513p1382
109	3p1382	513p1381
109	65p1822	513p1823
109	65p1823	513p1822
603	3p1381	3p1382
1459	513p1381	513p1382
513p1822	513p1823	514p572

---

HP(X(110),X(110)) passes through these centers:
1, 110, 9p1019, 21p1399, 35p109, 35p513, 655p2290

Collinear triples

1	110	21p1399
1	9p1019	35p513
110	9p1019	35p109
35p109	35p513	655p2290

---

HP(X(162),X(162)) passes through these centers:
1, 162, 656, 1822, 1823, 110p1312, 110p1313, 523p1113, 523p1114

Collinear triples

1	1822	1823
1	110p1312	523p1113
1	110p1313	523p1114
162	1822	523p1114
162	1823	523p1113
162	110p1316	110p1313
656	1822	110p1312
656	1823	110p1313
656	523p1113	523p1114

---

HP(X(163),X(163)) passes through these centers:
1, 163, 810, 3p32, 3p1379, 3p1380, 512p1078, 512p1379, 512p1380

Collinear triples

1	163	3p32
1	810	512p1078
163	3p1379	512p1380
163	3p1980	512p1379
810	512p1379	512p1380
3p32	3p1379	3p1380

---

HP(X(171),X(171)) passes through these centers:
1, 171, 894, 3p444, 43p894, 58p894, 63p444, 72p444

Collinear triples

1	43p894	72p444
171	894	3p444
171	58p894	72p444
894	43p894	63p444
3p444	63p444	72p444

---

HP(X(174),X(174)) passes through these centers:
1, 173, 174, 177, 236, 3p2091, 6p178, 177p258

Collinear triples

1	173	6p178
1	174	177
173	174	177p258
173	177	236
174	236	3p2091
3p2091	6p178	177p258

---

HP(X(189),X(189)) passes through these centers:
1, 4, 57, 1422, 2262, 2270, 4p282, 6p309, 63p1856, 77p2262, 84p946, 280p2262

Collinear triples

1	2262	2270
1	4p282	280p2262
1	6p309	84p946
4	57	84p946
4	2262	4p282
4	2270	63p1856
57	1422	2262
57	2270	77p2262
57	6p309	280p2262
1422	6p309	63p1856
4p282	6p309	77p2262
63p1856	84p946	280p2262

---

HP(X(220),X(9)) passes through these centers:
7, 9, 480, 9p1200, 218p220

Collinear triple

380

9p1200

218p220

---

HP(X(226),X(226)) passes through these centers:
1, 37, 65, 226, 442, 1781, 2294, 7p2294, 10p2160, 12p942, 73p445, 500p502

Collinear triples

1	65	73p445
1	226	12p942
1	1781	2294
37	65	2294
37	226	7p2294
37	1781	500p502
37	10p2160	12p942
65	226	442
226	10p2160	73p445
442	1781	10p2160
442	7p2294	12p942
2294	7p2294	73p445
12p942	73p445	500p502

---

HP(X(271),X(271)) passes through these centers:
1, 3, 271, 1433, 3p1854, 9p1073, 78p1854, 271p1854

Collinear triples

3	271	271p1854
3	1433	3p1854
271	1433	78p1854

---

HP(X(326),X(326)) passes through these centers:
1, 326, 2p1498, 3p1899, 69p426, 426p1032

Collinear triples

1	326	69p426
1	2p1498	3p1899
326	2p1498	426p1032

---

HP(X(329),X(329)) passes through these centers:
1, 196, 329, 2324, 40p1108, 196p1490, 223p1210, 1103p1210

Collinear triples

1	329	1103p1210
196	40p1108	196p1490
329	2324	223p1210

---

HP(X(333),X(333)) passes through these centers:
1, 8, 9, 283, 284, 333, 573, 2269, 2p960, 4p960, 21p960, 21p1193, 60p1211, 386p2339, 572p960

Collinear triples

1	8	2p960
1	9	572p960
1	283	60p1211
1	333	21p960
1	573	2269
8	9	2269
8	333	21p1193
9	284	21p960
9	333	60p1211
9	573	4p960
9	2p960	386p2339
283	284	2269
283	333	4p960
283	21p1193	386p2339
284	333	2p960
284	573	21p1193
2269	2p960	60p1211
2269	21p1193	572p960
2p960	4p960	21p960
21p960	21p1193	60p1211

---

HP(X(365),X(365)) passes through these centers:
1, 365, 3p367, 365p367

Collinear triple

1

365

6p367

---

HP(X(366),X(366)) passes through these centers:
1, 364, 366, 367, 366p367

Collinear triples

1	364	367
1	366	366p367

---

HP(X(385),X(385)) passes through these centers:
1, 385, 10p1580, 10p1691, 238p1580, 239p1966, 385p1284, 1281p1691 (no collinearities found)

---

HP(X(394),X(394)) passes through these centers:
1, 394, 836, 3p71, 3p72, 81p836, 86p836, 394p836

Collinear triples

1	394	394p836
394	3p71	86p836
394	3p72	81p836
836	3p71	3p72
81p836	86p836	394p836

---

HP(X(484),X(484)) passes through these centers:
1, 15p484, 16p484, 36p484, 58p484, 106p484, 203p484, 214p484, 484p501, 484p758, 484p1130

Collinear triples

1	214p484	484p501
15p484	16p484	58p484
16p484	36p484	203p484
36p484	58p484	484p501
36p484	214p484	484p758

---

HP(X(513),X(513)) passes through these centers:
1, 244, 513, 764, 1054

Collinear triples

1	244	1054
1	513	764

---

HP(X(515),X(515)) passes through these centers:
1, 515, 1455, 40p1455, 80p1455, 515p517, 515p1457, 517p1359

Collinear triples

1	515	517p1359
515	1455	515p517
1455	40p1455	515p1457
1455	80p1455	517p1359

---

HP(X(651),X(651)) passes through these centers:
1, 650, 651, 9p1381, 9p1382, 514p1381, 514p1382

Collinear triples

1	9p1381	9p1382
650	514p1381	514p1382
651	9p1381	514p1382
651	9p1382	514p1381

---

HP(X(662),X(662)) passes through these centers:
1, 662, 2p1379, 2p1380, 6p1113, 6p1114, 110p1312, 110p1313, 313p1919, 523p1380, 525p1113, 525p1114

Collinear triples

1	2p1379	2p1380
1	6p1113	6p1114
1	110p1312	525p1113
1	110p1313	525p1114
662	2p1379	523p1380
662	6p1113	525p1114
662	6p1114	525p1113
662	110p1312	110p1313
6p1113	110p1312	313p1919
6p1114	110p1313	313p1919
313p1919	525p1113	525p1114

---

HP(X(664),X(664)) passes through these centers:
1, 514, 664, 7p1376, 513p1376, 651p1376, 693p1037, 1376p1633

Collinear triples

1	514	513p1376
1	664	7p1376
514	664	651p1376
514	7p1376	693p1037
664	693p1037	1376p1633
513p1376	651p1376	1376p1633

---

HP(X(666),X(666)) passes through these centers:
1, 239, 294, 644, 666, 885, 9p659, 31p874, 105p812, 190p2348, 239p294, 740p919

Collinear triples

1	239	190p2348
1	294	740p919
1	666	239p294
1	885	105p812
239	294	666
239	9p659	105p812
239	31p874	239p294
294	644	239p294
294	885	9p659
644	666	105p812
644	9p659	190p2348
666	885	31p874
9p659	31p874	740p919

---

HP(X(765),X(765)) passes through these centers:
1, 100, 765, 3p1862, 1331p1862

Collinear triples

1	100	3p1862
100	765	1331p1862

---

HP(X(775),X(775)) passes through these centers:
1, 255, 775, 3p1885, 1105p1885

Collinear triples

1	255	3p1885
255	775	1105p1885

---

HP(X(811),X(811)) passes through these centers:
1, 811, 1577, 4p1975, 99p1968, 512p1975

Collinear triples

1	811	4p1975
1	1577	512p1975
811	1577	99p1968

---

HP(X(813),X(813)) passes through these centers:
1, 811, 1577, 4p1975, 99p1968, 512p1975

Collinear triples

1	813	292p2223
1	2284	31p1575
1	10p1922	291p2284
667	813	291p2284
667	875	292p2223
667	6p665	31p1575
813	875	2284
813	10p1922	21p1458
875	6p665	10p1922
2284	6p665	291p2284
2284	21p1458	292p2223

---

HP(X(894),X(894)) passes through these centers:
1, 171, 894, 2329, 58p1920, 171p1107, 894p1107, 1197p1215

Collinear triples

1	2329	1197p1215
171	894	894p1107
171	2329	171p1107
894	58p1920	1197p1215
171p1107	894p1107	1197p1215

---

HP(X(901),X(901)) passes through these centers:
1, 36, 106, 109, 901, 2p1983, 6p1019, 6p1022, 36p106, 36p513, 59p1769, 88p654

Collinear triples

1	36	59p1769
1	901	36p106
1	6p1022	88p654
36	106	901
36	2p1983	36p106
36	36p513	88p654
106	109	36p106
106	6p1022	36p513
109	901	88p654
109	36p513	59p1769
901	2p1983	6p1022
6p1019	6p1022	36p106

---

HP(X(908),X(908)) passes through these centers:
1, 119, 517, 908, 6p119, 6p153, 8p2183, 57p119, 80p1465, 119p517, 119p2323

Collinear triples

1	908	119p517
1	6p119	6p153
1	8p2183	119p2323
119	517	908
119	6p119	119p2323
119	57p119	119p517
517	6p119	8p2183
517	80p1465	119p517
908	8p2183	57p119
908	80p1465	119p2323
6p153	57p119	80p1465

---

HP(X(920),X(920)) passes through these centers:
1, 155, 920, 3p155, 3p454, 46p155, 90p155, 155p371, 155p372, 155p485, 155p487, 155p488, 493p1609, 494p1609, 920p1124, 920p1335

Collinear triples

1	920	3p454
155	3p155	46p155
155	3p454	90p155
155	155p371	920p1124
155	155p372	920p1335
920	46p155	90p155
920	155p371	155p485
920	155p372	155p486
920	155p487	494p1609
920	155p488	493p1609
920	920p1124	920p1335
3p155	155p371	155p372
3p155	155p487	155p488
3p454	155p485	155p486
3p454	493p1609	494p1609
46p155	155p485	920p1335
46p155	155p486	920p1124
155p371	155p486	155p487
155p372	155p485	155p488

---

HP(X(1010),X(1010)) passes through these centers:
1, 388, 1010, 1038, 388p958, 388p960, 940p2303, 1010p1193

Collinear triples

388	388p958	388p960
1010	388p958	1010p1193
1010	388p960	940p2303
1038	940p2303	1010p1193

---

HP(X(1025),X(1025)) passes through these centers:
1, 1025, 2283, 2284, 103p241, 241p926, 241p2254, 518p1362, 518p2254, 672p1280, 918p1381, 918p1382, 1381p2254, 1382p2254

Collinear triples

1	1025	518p1362
1025	2283	518p2254
1025	2284	241p2254
2283	2284	241p926
2283	103p241	518p1362
2284	518p1362	672p1280
241p926	918p1381	918p1382
241p2254	518p1362	518p2254
241p2254	1381p2254	1382p2254

---

HP(X(1150),X(1150)) passes through these centers:
1, 1150, 10p993, 10p2278, 993p995, 995p1150

Collinear triples

1150	10p993	993p995
1150	10p2278	995p1150

---

HP(X(1171),X(1171)) passes through these centers:
1, 6, 58, 1171, 37p1126, 37p1203, 81p1203, 1126p1203, 1203p1255

Collinear triples

1	58	81p1203
1	1171	1126p1203
6	58	37p1203
6	1171	1203p1255
6	37p1126	1126p1203
1171	37p1126	81p1203
37p1203	81p1203	1203p1255

---

HP(X(1309),X(1309)) passes through these centers:
1, 104, 108, 1309, 104p522, 108p515 (no collinearities found)

HP(X(1381),X(1381)) passes through these centers:
1, 56, 266, 1381, 513p1382

Collinear triple

1

56

1381

---

HP(X(1400),X(1400)) passes through these centers:
1, 213, 1400, 1402, 2092, 6p2092, 37p904, 56p2292, 73p444, 181p2300

Collinear triples

1	1400	181p2300
1	1402	73p444
213	1400	56p2292
213	1407	6p2092
213	37p901	181p2300
1400	1402	2092
1400	37p904	73p444
2092	56p2292	181p2300
6p2092	56p2292	73p444

---

HP(X(1423),X(1423)) passes through these centers:
1, 1403, 1423, 2176, 43p982, 43p2275, 982p1423

Collinear triples

1403	1423	43p982
1403	2176	43p2275
1423	2176	982p1425

---

HP(X(1445),X(1445)) passes through these centers:
1, 218, 1445, 1617, 9p218, 55p218, 218p218

Collinear triples

1	1445	218p218
218	1617	55p218
218	9p218	218p218
1445	1617	9p218

---

HP(X(1708),X(1708)) passes through these centers:
1, 1708, 3p1708, 6p1708, 35p1780, 37p1780, 71p1780, 72p1780

Collinear triples

1708	3p1708	37p1780
1708	6p1708	72p1780
3p1708	6p1708	72p1780
6p1708	37p1780	71p1780

---

HP(X(1737),X(1737)) passes through these centers:
1, 119, 912, 1737, 3p119, 36p119, 46p912, 80p912, 90p119, 119p912

Collinear triples

1	1737	119p912
1	36p119	46p912
119	912	1737
119	3p119	36p119
119	90p119	119p912
912	3p119	46p912
912	80p912	119p912
1737	36p119	80p912
1737	46p912	90p119

---

HP(X(1740),X(1740)) passes through these centers:
1, 194, 1740, 2p194, 43p194, 87p194, 194p194

Collinear triples

1	1740	194p194
194	2p194	43p194
194	87p194	194p194
1740	43p194	87p194

---

HP(X(1748),X(1748)) passes through these centers:
1, 563, 1748, 2p571, 2p1147, 4p571, 24p1147, 155p1993, 254p571

Collinear triples

1	1748	24p1147
1	2p571	155p1993
563	2p571	4p571
563	2p1147	155p1993
1748	2p1147	4p571
1748	155p1993	254p571
2p571	2p1147	24p1147
4p571	24p1147	254p571

---

HP(X(1749),X(1749)) passes through these centers:
1, 1749, 3p1157, 35p1749, 36p1749, 195p1157, 1094p1749, 1095p1749, 1154p1157

Collinear triples

1749	35p1749	36p1749
3p1157	195p1157	1154p1157

---

HP(X(1760),X(1760)) passes through these centers:
1, 3p22, 4p206, 6p206, 22p206, 22p1677, 206p1670, 206p1671

Collinear triple

6p206

206p1670

206p1671

---

HP(X(1761),X(1761)) passes through these centers:
1, 1761, 3p199, 27p199, 58p199, 58p1330, 199p2248

Collinear triple

1761

3p199

27p199

---

HP(X(1762),X(1762)) passes through these centers:
1, 1762, 21p1762, 31p1762, 48p1762, 1172p1762, 1762p2194

Collinear triples

1762	21p1762	31p1762
1762	48p1762	1172p1762
31p1762	48p1762	1762p2194

---

HP(X(1953),X(1953)) passes through these centers:
1, 1953, 6p233, 17p51, 18p51, 51p233, 62p233

Collinear triples

1	1953	51p233
1953	18p51	62p233
17p51	18p51	51p233

---

HP(X(1958),X(1958)) passes through these centers:
1, 1957, 1958, 2p1975, 20p1968, 194p1975, 1196p1975

Collinear triples

1957	20p1968	1196p1975
2p1957	194p1975	1196p1975

---

HP(X(2155),X(2155)) passes through these centers:
1, 1973, 2155, 6p800, 19p2155, 64p185

Collinear triples

1973	2155	64p185
1973	6p800	19p2155

---

HP(X(2160),X(2160)) passes through these centers:
1, 2160, 2308, 6p1251, 41p554, 554p2308, 1081p2308, 2160p2308

Collinear triples

1	2160	2160p2308
2160	6p1251	554p2308
2160	41p554	1081p2308
2308	554p2308	1081p2308
6p1251	41p554	2160p2308

---

HP(X(2166),X(2166)) passes through these centers:
1, 1725, 1749, 2166, 265p403

Collinear triples

1	1725	1749
1	2166	265p403

---

HP(X(2222),X(2222)) passes through these centers:
1, 513, 859, 1319, 1769, 2222, 10p1411, 11p2222, 59p1769, 80p1457, 517p1168, 517p2222

Collinear triples

1	1319	59p1769
1	2222	80p1457
1	10p1411	517p2222
513	1319	1769
513	2222	517p2222
513	11p2222	80p1457
859	2222	10p1411
859	59p1769	80p2222
1319	2222	517p1168
1769	10p1411	11p2222
1769	59p1769	517p2222
2222	11p2222	59p1769
80p1457	517p1168	517p2222

---

HP(X(2285),X(2285)) passes through these centers:
1, 1460, 2285, 6p612, 171p1010, 388p1468, 388p2268, 612p1468, 1999p2303

Collinear triples

1460	2285	388p2268
1460	6p612	612p1468
1460	171p1010	388p1468
2285	6p612	388p1468
6p612	388p2268	1999p2303
171p1010	612p1468	1999p2303

---

HP(X(2287),X(2287)) passes through these centers:
1, 200, 220, 2287, 2p2332, 9p2264, 20p2332, 21p950, 21p2264, 78p2264

Collinear triples

1	220	78p2264
1	20p2332	21p2264
200	220	9p2264
200	2287	21p650
220	2287	21p950
2287	2p2332	78p2264
9p2264	9p2264	20p2332
21p650	21p2264	78p2264

---

HP(X(2311),X(2311)) passes through these centers:
1, 41, 2311, 8p237, 9p1967, 21p2223, 240p1808

Collinear triples

41	2311	240p1808
41	8p237	21p2223

---

HP(X(2316),X(2316)) passes through these centers:
1, 2316, 9p2183, 15p1251, 1833p2193

Collinear triple

9p2183

15p1251

1833p2193

---

HP(X(2319),X(2319)) passes through these centers:
1, 8, 2319, 2p2319, 9p2275, 41p982, 2053p2275

Collinear triples

8	2319	2053p2275
8	2p2319	9p2275
2319	2p2319	41p982

---