points and transformations on conics
I came up with a nice formula that, given a point on a circumconic (perspector P) gives the corresponding point on its dual, the inconic with perspector tP.
The dual of a point on an circumconic, pespector P, is the tangent line to an inconic, perspector tP. This leads to a nice formula relating a point on a circumconic to the corresponding point (the point of tangency) on its dual.
Let the circumconic, perspector P = (l:m:n) have equation ... + m/y + .... = 0
Its corresponding inconic has equation .... + root( my )+ ... = 0
If Q = ( : y1 : ) is on this conic then ... + m/y1 + .... = 0.
The dual of Q is ... + y1 y + ... = 0
The point : m/y1^2 : is manifestly on this line as well as the inconic. So this must be the point of tangency.
Hence the transformation : y : -> : m/y^2 : takes a point on a circumconic to the correcponding point of tangency on its dual inconic.
As an aside : m/y : is on the line at infinity.
Example: Kiepert hyperbola (perspetor : cc - aa : = tS) and its dual, the Kiepert parabola (perspector S)
Point on KH Point on KP
G : cc - aa :, its infinite point
H : SBB (cc-aa) :
R = tK : aaaa(cc - aa) :
So : (c-a)/(c+a) :
1/(SB + constant) : (SB + const)^2 (cc-aa) : this is general point
1/(aa + constant) : (aa + const)^2 (cc-aa) : another way to write general point
[Peter Moses] Dear Steve,
P{p1, p2, p3} on L.inf to a circumconic with perspector U{u1, u2, u3}
Inf2Circumconic[{u1_, u2_, u3_}, {p1_, p2_, p3_}] := {u1 / p1, u2 / p2, u3 / p3}
P{p1, p2, p3} on L.inf to a inconic with perspector U{u1, u2, u3}
Inf2Inconic[{u1_, u2_, u3_}, {p1_, p2_, p3_}] :=
{p1^2 u1, p2^2 u2, p3^2 u3}
An easy example is
P{p1, p2, p3} on L.inf to Steiner circumellipse (perspector G) =
{1 / p1, 1 / p2, 1 / p3}, the isotomic conjugate of P.
Another example
P{p1, p2, p3} on L.inf to Orthic inconic (center K, perspector H) =
{p1^2 / SA, p2^2 / SB, p3^2 / SC}
In this example X(514) -> X(2969) and X(525) -> X(125), currently
the only 2 ETC points on the Orthic inconic.
I reckon on these ETC points being on L.inf ...
{30,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,
527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,
544,545,674,680,688,690,696,698,700,702,704,706,708,710,712,714,716,
718,720,722,724,726,730,732,734,736,740,742,744,746,752,754,758,760,
766,768,772,776,778,780,782,784,786,788,790,792,794,796,802,804,806,
808,812,814,816,818,824,826,830,832,834,838,888,891,900,912,916,918,
924,926,928,952,971,1154,1499,1503,1510,1912,1938,1946,2385,2386,
2387,2388,2389,2390,2391,2392,2393,2574,2575,2771,2772,2773,2774,
2775,2776,2777,2778,2779,2780,2781,2782,2783,2784,2785,2786,2787,
2788,2789,2790,2791,2792,2793,2794,2795,2796,2797,2798,2799,2800,
2801,2802,2803,2804,2805,2806,2807,2808,2809,2810,2811,2812,2813,
2814,2815,2816,2817,2818,2819,2820,2821,2822,2823,2824,2825,2826,
2827,2828,2829,2830,2831,2832,2833,2834,2835,2836,2837,2838,2839,
2840,2841,2842,2843,2844,2845,2846,2847,2848,2849,2850,2851,2852,
2853,2854,2869,2870,2871,2872,2873,2874,2875,2876,2877,2878,2879,
2880,2881,2882}
Best regards,
Peter.
Edward's page
http://pages.infinit.net/spqrsncf/ngorecent.htm
and there is MathWorld.
But to go on with, here are a few in one place ...
There are bound to be some I have missed. Some are not listed because they
have very few or no ETC points ..e.g. the Adams circle
Any that I should add, let me know.
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For a point P on some Circumconic with perspector U, P V / U is on another Circumconic with perspector V.
For a point P on some Inconic with perspector U, P V / U is on another Inconic with perspector V.
For a point P on some line l1 x + l2 y + l3 z = 0, V / (L P) is on a Circumconic with perspector V.
For a point P on some line l1 x + l2 y + l3 z = 0, L^2 P^2 V is on an Inconic with perspector V.
For a point P on some Circumconic with perspector U, U^2 V / P^2 is on an Inconic with perspector V.
For a point P on some Inconic with perspector U, V Sqrt[U / P] is on a Circumconic with perspector V.
p y z + q z x + r x y = 0 is a circumconic with perspector P{p,q,r} and center p(p - q - r) :: (G-Ceva Conjugate P)
x^2 / p^2 - 2 y z / (q r) + cyclic is an inconic with perspector P{p,q,r} and center p (q + r) :: (G-CrossPoint P) = complement of the isotomic of P. If P is on the Steiner circumellipse, the "inconic" is a parabola with a focus on the CC at isog[isot[P]]
Dual of a Circumconic with perspector U is an Inconic with perspector isot[U]
Dual of an Inconic with perspector U is a Circumconic with perspector isot[U]
Circumconics with perspectors P{p,q,r} & U{u,v,w} intersect at A,B,C & {1 / (p3 u2 - p2 u3)::}, the Isogonal of the Cross Difference
--------------------------------------------------------------------------------------------------------
Circumcircle, center 3, perspector 6
{74,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,476,477,675,681,689,691697,699,701,703,705,707,709,711,713,715,717,719,721,723,725,727,729,731,733,735,737,739,741,743,745,747,753,755,759,761,767,769,773,777,779,781,783,785,787,789,791,793,795,797,803,805,807,809,813,815,817,819,825,827,831,833,835,839,840,841,842,843,898,901,907,915,917,919,925,927,929,930,931,932,933,934,935,953,972,1113,1114,1141,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1379,1380,1381,1382,1477,2222,2249,2291,2365,2366,2367,2368,2369,2370,2371,2372,2373,2374,2375,2376,2377,2378,2379,2380,2381,2382,2383,2384,2687,2688,2689,2690,2691,2692,2693,2694,2695,2696,2697,2698,2699,2700,2701,2702,2703,2704,2705,2706,2707,2708,2709,2710,2711,2712,2713,2714,2715,2716,2717,2718,2719,2720,2721,2722,2723,2724,2725,2726,2727,2728,2729,2730,2731,2732,2733,2734,2735,2736,2737,2738,2739,2740,2741,2742,2743,2744,2745,2746,2747,2748,2749,2750,2751,2752,2753,2754,2755,2756,2757,2758,2759,2760,2761,2762,2763,2764,2765,2766,2767,2768,2769,2770,2855,2856,2857,2858,2859,2860,2861,2862,2863,2864,2865,2866,2867,2868,3067}
Anticomplement of Circumcircle,C(X(4),2 R), center 4, perspector 4
{146,147,148,149,150,151,152,153}
Incircle, center 1, perspector 7{11,1314,1315,1317,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,2446,2447,3021,3022,3023,3024,3025,3026,3027,3028}
NP circle, center 5, perspector {4,160} /\ {5,141}{11,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,1312,1313,1560,1566,2039,2040,2679}
Brocard circle, center 182, perspector a^2/(2 a^4 + b^2 c^2) ::, isog[{2,187}/\{3,83}/\{6,99}/\{32,76}]
{3,6,1083,1316}
Parry circle, center 351
{2,15,16,23,110,111,352,353}
Complement of the Parry circle, center (b^2 - c^2) (b^4 + c^4 - 5 b^2 c^2 + 4 S^2) :: = {2,351}/\{125,526}/\{512,625}
{2,125,126,623,624,858}
Lester circle, center 1116
{3,5,13,14,1117}
Cosine circle, center 6, perspector a^2/(S^2 + a^2 SA) :: = {3,51}/\{5,69}/\{25,54}
{1666,1667}
Gallatly circle, center 39, perspector isog [{2,98}/\{3,83}/\{110,114}]
{2026,2027}
Orthocentroidal circle, center 381, perspector isog[{2,98}/\{6,23}/\{20,54}]
{2,4}
Lemoine circle, center 182, perspector isog[{2,39}/\{3,83}]
{1662,1663}
Spieker circle, center 10, perspector 2
{3035,3036,3037,3038,3039,3040,3041,3042}
Bevan circle, center 40, perspector 57
{1054,1282,1768,2100,2101,2448,2449,2948}
Sine Triple angle circle, center 49, perspector isog[{3,93},{94,184}]
{3043,3044,3045,3046,3047,3048}
Apollonius circle, center 970, perspector isog[{2,261}/\{312,993}/\{333,572}/\{1078,1211}]
{2037,2038,3029,3030,3031,3032,3033,3034}
Fuhrmann circle, center 355
{4,8}
Mandart circle, center 1158
{11,1364}
Tangential circle, center 26, perspector 1485
{2079,2931}
Evans circle, center 1019
{1,484,1276,1277}
Jerabek circum-hyperbola, center 125, perspector 647 {3,4,6,54 (gN) ,64 (gL) , 65 (gHo), 66 (sH), 67 (gvG), 68 (g24), 69 D, 70 ,71,72,73 (g29),74 (g einf), 248, 265 (tvH), 290, 695 (gK2+), 879,895,1173,1175, 1176,(sHo) 1177, 1242,1243,1244,1245,1246,1439,1798,1903,1942,1987,2213,2435,2574,2575,2992,2993}
X(54) = KOSNITA POINT = gN
X(64) = ISOGONAL CONJUGATE OF X(20)=bb/(bbB-CA)
X(71) = ISOGONAL CONJUGATE OF X(27) = bb (c + a) SB
X(72) = ISOGONAL CONJUGATE OF X(28) = b(c + a) SB
X(895) = ISOGONAL CONJUGATE OF X(468) = a2(b2 + c2 - a2)/(b2 + c2 - 2a2)
Kiepert circum-hyperbola, center 115, perspector 523
{2,4,10,13,14,17,18,76 (R), 83 (tmK), 94, 96, 98, 226, 262,275, 321, 485,486 (Vecten),598 ( (1/(2c2–b2+2a2)), 671 ( (1/(c2–2b2+a2), 801,1029,1131,1132,1139, 1140, 1327 ((b(c-a)/(c2+a2–bc–ab)), 1328, 1446,1676,1677,1751, 1916, 2009,2010,2051, 2052 (sec 2B), 2394,2592,2593,2671,2672,2986,2996} purple = centers of similiarity
Feuerbach circum-hyperbola, center 11, perspector 650{1,4,7,8,9,21,79,80,84,90,104,177,256,294,314,885,941,943,981,983,987,989,1000,1039,1041,1061,1063,1156,1172,1251,1320,1389,1392,1476,1896,1937,2298,2320,2335,2344,2346,2481,2648,2997,3062,3065}
A,B,C,H,N. center 1141
{4,5,53,311,327,1141,1263,1487,2165,2980}
A,B,C,H,X(12) center on NP circle
{4,12,442,1234,1865}
A,B,C,H,X(19), center 915
{4,19,28,34,286,915,1118,1119}
A,B,C,H,X(25), center (b^2 - c^2)^2 (a^2 - SA) SB SC ::{4,99} /\ {132,235}, perspector 2489
{4,25,683,1426,1824,2207,2333}
A,B,C,H,X(40), center {9,119} /\ {10,118} /\ {12,208}
{4,40,57,189,196,223,329,937,972,1817,2184}
A,B,C,H,X(56), center {2,901} /\ {4,953} /\ {11,513}
{4,56,513,517,859,945,953,957,1457,1875,2183}
A,B,C,H,X(75), center (b - c)^2 (b^2 + c^2 + a b + b c + c a) (a^2 + b^2 + c^2 + 2 b c) :: {2,835}/\{116,244}/\{121,1054}/\{125,1086}
{4,75,388,1010,1065,1220,2345}
A,B,C,H,X(93), center 136, perspector 2501
{4,93,225,254,264,393,847,1093,1105,1179,1217,1300,1826}
A,B,C,G,K. center 1084, perspector 512{2,6,25,37,42,111,251,263,308,393,493,494,588,589,694,941,967,1169,1171,1218,1239,1241,1383,1400,1427,1880,1976,1989,2054,2165,2248,2350,2395,2433,2963,2981,2987,2998}
A,B,C,G,Ge. center 1086, perspector 514 {2,7,27,75,86,234,272,273,310,335,554,673,675,871,903,1081,1088,1223,1240,1246,1268,1440,1659,2296,2400,2989}
A,B,C,G,Na. center 1146, perspector 522
{2,8,29,85,92,178,189,257,312,333,1121,1220,1311,1952,2090,2399,2988,2994}
A,B,C,G,O. center a^4 (b^2 - c^2)^2 SA^4 :: {3,1625}/\{115,122}/\{216,549}, perspector 520
{2,3,97,276,394,1073,1214,1217,1297,2416}
A,B,C,I,G. center 1015, perspector 513{1,2,28,57,81,88,89,105,274,277,278,279,291,330,367,955,957,959,961,985,1002,1022,1123,1170,1219,1224,1255,1257,1258,1280,1336,1390,1422,1432,1929,2006,2224,2282,2306,2362,2401,2982,2990}
A,B,C,I,O. center longish, perspector 652{1,3,29,77,78,102,219,282,283,284,296,332,945,947,949,951,1036,1037,1057,1059,1065,1067,1069,1433,1794,1795,1807,2066,2338,2359,2656}
A,B,C,I,K. center a^2 (b - c)^2 (a^2 + a b + c a - b c) ::, perspector 649{1,6,34,56,58,86,87,106,269,292,870,937,939,977,979,996,998,1027,1120,1126,1167,1220,1222,1256,1411,1413,1431,1438,1474,2163,2191,2215,2279,2297,2334,2336,2424,2665,2983}
A,B,C,N,K. center long
{5,6,24,52,847,1166,2383}
Dual Keipert = Kiepert parabola, "center" 523, perspector 99
{523,669,1649,2528}
Dual circumcircle, center 141, perspector 76
{338,1086}
Dual Incircle, center (b + c - a) (b + c - 3 a) :: {7,190}/\{8,9}. , perspector 8
{190,645,646}
Steiner rectangular hyperbola, center 99
{1,2,20,63,147,194,487,488,616,617,627,628,1670,1671,1764,2128,2582,2583,2896}
Feuerbach tangential = Stammler hyperbola, center 110 {1,3,6,155,159,195,399,610,1498,1740,2574,2575,2916,2917,2918,2929,2930,2931,2935,2948}
Kiepert,tangential
{154,155,184}
Jerabek,tangential
{4,26,155,157,2165}
Kiepert,excentral, center 101
{1,43,165,170,365,846,1051,1282,2108,2536,2537,2939,2944,2947,2954}
Jerabek,excentral, center 100
{1,9,40,188,191,366,1045,1050,1490,2136,2949,2950,2951}
Feuerbach,excentral
{1,164,166,167,168}
Kiepert,orthic
{51,52,129,389}
Jerabek,orthic
{5,52,53,128,570}
Feuerbach,orthic .. Kimberling, center 1112, perspector 1/((-a^2 + b^2 + c^2) (a^4 - a^2 b^2 + 2 b^4 - a^2 c^2 - 3 b^2 c^2 + 2 c^4)) :: isot[{69,125}/\{287,343}]
{4,6,52,113,155,185,193,1162,1163,1829,1839,1843,1858,1986,2574,2575,2904,2905,2906,2907,2914}
Kiepert,intouch
{65,354,1362,1401}
Jerabek,intouch
{1,7,65,145,224,1071,1317,1537}
Keipert,anticomp, center 99
{1,2,20,63,147,194,487,488,616,617,627,628,1670,1671,1764,2128,2582,2583,2896}
Feuerbach,anticomp, center 100
{7,8,20,144,145,153,2475,2894,2897}
Jerabek,anticomp, center 110
{4,20,69,146,193,2574,2575,2888,2889,2892}
Steiner circumellipse, center 2, perspector 2
{99,190,290,648,664,666,668,670,671,886,889,892,903,1121,1494,2479,2480,2481,2966}
MacBeath circumconic, center 6, perspector 3 {110,287,648,651,677,895,1331,1332,1797,1813,1814,1815,2133,2986,2987,2988,2989,2990,2991}
Steiner inellipse, center 2, perspector 2
{115,1015,1084,1086,1146,2454,2455,2482}
Brocard inellipse, center 39, perspector 6, (axis Brocard axis, foci Brocard points)
{1015,1017,1977,2028,2029}
dual Brocard inellipse, center b^2 c^2 (a^2 b^2 + c^2 a^2 - b^2 c^2) ::, perspector 76
{670,689,1978}
Orthic inconic, center 6, perspector 4, (axis {6, 1344, 2574}, axis {6, 1345, 2575})
{125,2969}
MacBeath inconic, center 5, perspector 264 (axis Euler line)
{339,1312,1313,2967,2968,2969,2970,2971,2972,2973,2974}
Yff Parabola, dual (A,B,C,G,Ge). "center" 514, perspector 190
{514,649}
de Longchamps ellipse, center 1, perspector a / (a b + b c + c a - a^2) :: = {10,141}/\{37,38}/\{81,82}
{244,2170,2611,2446,2447}
dual of Mandart inellipse, center {1,7} /\ {3,934} /\ {8,348} /\ {55,479},
{658,664,927}
Mandart hyperbola, center {8,80} /\ {11,210} /\ {63,100}
{8,9,40,72,144,1145,3057,3059}
dual Mandart parabola, center 664
{2,145,174,175,176,188,508}
Lemoine inconic, center 597, perspector 598 (axis GK line)
Mandart inconic, center 9, perspector 8 (axis {9,2590}, axis {9,2591})
{11}
incentral inconic, center 37, perspector 1
{244,678,2310,2632,2638,2643}
incentral circumconic, center 9, perspector 1{88,100,162,190,651,653,655,658,660,662,673,771,799,823,897,1156,1492,1821,2349,2580,2581}
orthic circumconic, center 1249, perspector 4
{107,648,653,685,687,1897}
NP circumconic, center 2165, perspector 5
{94,648,655,925,1972}
Mittenpunkt circumconic, center 1, perspector 9
{100,643,644,664,1120,1280,1320,1897}
Feuerbach circumconic, center 650, perspector 11
{514,522,655,666,885,929,2401}
Clawson circumconic, center a SB SC (a SB SC - b SC SA - c SA SB)::, perspector 19
{108,162,811,1783,1897,2586,2587}
2rd power point circumconic, center a^3 (a^3 - b^3 - c^3), perspector 31
{101,163,662,692,909,911,913,923,1415,1438,1461,1910,2159,2224,2576,2577}
3rd power point circumconic, center 206, perspector 32
{110,685,692,1177,1492,1576,1976}
37 circumconic, center 10, perspector 37
{80,100,291,668,1018,1783}
Brocard midpoint circumconic, center 141, perspector 39
{67,110,660,670,694,1634}
42 circumconic, center a^2 (b + c) (a^2 b - a b^2 + a^2 c - b^2 c - a c^2 - b c^2) ::, perspector 42
{101,190,292,1018,1020,2161,2250}
48 circumconic, center a^3 SA (a^3 SA - b^3 SB - c^3 SC)::, perspector 48
{109,162,163,293,906,1331,1795,1822,1823}
55 circumconic, center a^2 sa (S (r + R) - a^2 sa)::, perspector 55
{101,294,644,645,651,666,1783,2311,2316,2338,2341}
63 circumconic, center a SA (a SA - b SB - c SC):: , perspector 63
{336,662,664,811,1310,1332}
115 circumconic, center , perspector 115
{476,523,685,850,892,2395,2501}
125 circumconic, center , perspector 125
{523,525,879,935,2394,2966}
184 circumconic, center , perspector 184
{112,248,906,1415,1576,2966}
(format below :- perspector X(n) {X points on conic}
219 {100,1331,1793,1808,1809,1810,1811}
244 {513,514,876,1019,1022,1027,1308}
521 {2,21,63,78,280,345,348,1791,1812,2339,2417}
525 {2,69,95,253,264,287,305,306,307,328,1441,1494,1799,1972,2373,2419}
526 {2,15,16,186,249,323,842,1138,2411}
654 {1,36,54,59,60,953,1318,1391,1443,1870,2323,2364,2597}
656 {1,63,72,92,226,293,304,306,1214,1956,2167,2184,2349,2582,2583}
657 {1,33,55,64,103,200,220,963,1043,2192,2328,2332,2342}
659 {1,83,238,239,1016,1019,1244,1429,1509,2111}
661 isog[{1,21}], center 244, hyperbola (isotomic of Io -- tIo ){1,10,19 (gWo),37, 65 (gHo),75 (tIo), 82 ( b/(c2 + a2)),91 (g47),158, 225 ,267, 596 ,759, 775, 876, 897, 921, 969, 994, 1247, 1581(a/(a4 - b2c2) ), 1910, 2153 (1/bFn),2154(1/bFs), 2166 (1/b(SBB-ccaa), 2168, 2186, 2190, 2214 (a/(cc+aa+S11), 2216, 2217, 2218, 2219, 2363 (a/(c+a)(cc+aa+ab+bc)),2588,2589, 2652, 2962}
a (8 S^2 – 6 a^2 SB + r3 a^2 Δ)
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CC(661) Meets inf at {a / (EE - FF ZZ) ::}
SEARCH -9.082261517916963 & 0.9552773909328346
where
ZZ = +/- Sqrt[(a + b) (b + c) (c + a) (a^2 b + a b^2 + a^2 c - 6 a b c + b^2 c + a c^2 + b c^2)]
EE = (a + b) (a + c) (2 a^4 - 2 a^3 b - a^2 b^2 + 2 a b^3 - b^4 - 2 a^3 c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 + 2 a c^3 - b c^3 - c^4) :: {2,6},{63,1019}
FF = (b + c) (a^2 - b^2 - c^2 + b c) :: {2,6},{8,442},{226,306},{514,661}
Both EE & FF are on the G -- K line
Even better for trying to plot in GSP is something like
a / (3 (b + c) (2 SA - b c) pp zz + (a + b) (c + a) ((2 a - b - c) qq - 2 s zz^2)) ::
pp = Sqrt[(a + b) (b + c) (c + a)]
qq = (3 a^3 + 3 b^3 + 3 c^3 - 9 a b c - zz^2) = 6(-2abc - 2sabc + so SW - zz^2)
zz = +/- Sqrt[(a^2 b + a b^2 + a^2 c - 6 a b c + b^2 c + a c^2 + b c^2)] =
zz^2 = -5abc + 4sabc + 2so SW
qq = 7 abc - 8 sabc + 4 so SW
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663 isog[{2,7}], center a^2 (a - b - c) (b - c)^2 (a^3 - 2 a^2 b + a b^2 - 2 a^2 c + a b c - b^2 c + a c^2 - b c^2) :: hyperbola {6,9,19,55,57,284,333,673,893,909,1024,1174,1436,1751,1945,2160,2161,2164,2195,2258,2259,2291,2299,2316,2319,2337,2339,2343,2364,2432,2590,2591}
665 {6,7,59,513,518,672,840,1002,1037,1458,1876,2113}
667 isog[{2,37}], center a^3 (b - c)^2 (a^3 + a b^2 + a b c - b^2 c + a c^2 - b c^2)::
{6,31,81,604,608,739,1333,1407,1462,1911,2162,2203,2214,2221,2255,2298,2423,2991}
669 {6,32,83,213,729,981,1918,1974,2207,2281,2422}
672 {1,101,664,1026,2283,2284}
684 {3,76,249,297,511,525,2710}
693 {75,76,85,274,286,331,334,767,870,1218,1221,2481}
764 {88,244,335,1086,1462,3020}
788 {2,31,292,743,869,893,2276}
798 isog[{1,75}] {1,31,42,213,741,875,923,1042,1096,1245,1402,1967,1973,2107,2258,2296}
810 isog[{19,27}] {19,31,48,63,71,228,1400,1409,1820,1821,2148,2155,2156,2157,2158,2159,2215,2249,2250,2281,2282,2357,2578,2579}
822 {1,48,73,255,326,336,821,1248,2169,2584,2585,2660}
826 {2,66,141,427,1031,1502}
850 {76,264,276,290,300,301,308,313,327,349,1502,2367}
900 {2,514,519,996,1000,1016,2726,2985}
905 {7,63,69,77,81,189,286,969,1444,1814,2995}
924 {2,24,54,254,371,372,1993}
926 {2,55,241,650,672,949,1252,2115,2340}
1021 {21,29,86,285,1043,1098,2287,2326}
1084 {512,669,805,875,881,886}
1459 {3,27,57,58,63,84,103,222,295,967,1790,1796,1797,1803,1810,2067,2221}
1510 {2,61,62,288,1166,1994,2413,2984}
1577 {75,92,313,321,561,1441,1821,1934,2995,2997}
1635 {1,44,513,519,679,751,765,1168,1319,1877,2718}
1637 {4,30,477,523,1138,1990}
1639 {8,519,522,596,2325,2757}
1640 {67,98,523,542,1494,1989}
1643 {105,513,528,1156,2161,2246}
1769 {57,92,514,517,908,994,998,1168,1243,1465,2051,2717}
1919 {31,32,58,727,985,987,1106,1395,1397,1416,1472,1922,2206}
1924 {31,82,560,715,1918,1927,2205}
1960 {6,44,89,649,902,1252,1404,2226,2384}
2081 {5,54,93,186,523,1154,1273,2594,2599}
2254 {1,85,241,514,518,2725}
2451 {64,98,1105,1968,1975,1988}
2488 {6,279,1212,1418,1475,2293}
2489 {4,25,683,1426,1824,2207,2333}
2491 {4,32,237,263,511,512,2211,2698}
inellipse 59, center (a - b)^2 sc / c^2 + (c - a)^2 sb / b^2 = {1,39}/\{9,216}
{55,56,181,202,203,215,1124,1335,1362,1397,1672,1673,1682,2007,2008}
inellipse 69, center X(3), dual of "orthic circumconic", center 1249, perspector 4
{125,1565,2968}
inconic 249 center (a^2 - b^2)^2 / c^2 + (c^2 - a^2)^2 / b^2 = {2,94}/\{141,216}
{6,394,593,1501,1599,1600}
inconic 1101 {31,255,849,1094,1095,1917}