# 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

## 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.

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

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

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

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.

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