[Antreas P. Hatzipolakis]:
Let ABC be a triangle and A'B'C' the cevian triangle of G
Denote:
Ab, Ac = (Perpendicular to BC at A' [ = Perpendicular bisector of BC]) Intersection BB', CC' resp,
Bc, Ba = (Perpendicular to CA at B' [ = Perpendicular bisector of CA]) Intersection CC', AA' resp.
Ca, Cb = (Perpendicular to AB at C' [ = Perpendicular bisector of AB]) Intersection AA', BB' resp.
The BcCb, CaAc, AbBa are concurrent. Point?
Loci:
A'B'C' = cevian triangle of a point P.
1.
Ab, Ac = (Perpendicular to BC at A' ) Intersection BB', CC' resp,
Bc, Ba = (Perpendicular to CA at B' ) Intersection CC', AA' resp.
Ca, Cb = (Perpendicular to AB at C' ) Intersection AA', BB' resp.
2.
Ab, Ac = ( Perpendicular bisector of BC) Intersection BB', CC' resp,
Bc, Ba = (Perpendicular bisector of CA) Intersection CC', AA' resp.
Ca, Cb = (Perpendicular bisector of AB) Intersection AA', BB' resp.
Which are the loci of P such that BcCb, CaAc, AbBa are concurrent ?
[César Lozada]:
1) Locus1 = {Altitudes} ∪ { Lucas cubic}
ETC pairs (P,Z1(P)=point of concurrence): (2,141), (7,226), (8,6736), (69,394)
Z1(X(20)) = (4*(-SW+4*R^2)*(SA-SW)-S^2)*( SA*(16*R^2-SW-3*SA)-2*S^2) : : (barycentrics)
= on Euler line {2,3}
= {X(20), X(6616)}-Harmonic conjugate of X(3)
= [ -79.223148446053390, -79.88367570299195, 95.509277712926530 ]
Z1( X(189) ) = ((b+c)*a-(b-c)^2)/a/(a^3+(b+c) *a^2-(b+c)^2*a-(b^2-c^2)*(b-c) ) : : (trilinears)
= 3*(r+2*R)*X(2)-(r+6*R)*X(77)
= On lines: {2,77}, {84,516}
= {X(189), X(1440)}-Harmonic conjugate of X(282)
= [ 97.729603559387160, 80.56476384415213, -97.240912129915000 ]
Z1( X(253) ) = ((b^2+c^2)*a^2-(b^2-c^2)^2)/ a/(3*a^4-2*(b^2+c^2)*a^2-(b^2- c^2)^2) : : (trilinears)
= on cubic K671 and lines: {2,253}, {5,8798}, {30,64}, {381,6526}, {8703,11589}
= [ -1.194173815751119, 0.48107119998090, 3.858772335344276 ]
2) The entire plane.
For P=u:v:w (trilinears), Z2(P)=point of intersection=
= (w*b+c*v)/a*u : :
= Inverse_in_NPC(Complement( Isogonal(AntigonalConjugate(P) )))
= Isogonal(Complement(Isotomic( CyclocevianConjugate( PolarConjugate(Isogonal( Isotomic(PolarConjugate(P))))) )))
ETC pairs (P,Z2(P)): Very long list (see below)
César Lozada
ETC pairs (P,Z2(P)):
(1,10), (2,141), (3,3), (4,5), (5,1209), (6,2), (7,2886), (8,1329), (9,3452), (10,3454), (13,623), (14,624), (15,618), (16,619), (17,635), (18,636), (19,226), (20,2883), (21,960), (22,206), (23,6593), (24,1147), (25,6), (28,942), (30,113), (31,37), (32,39), (34,1210), (35,3647), (36,214), (37,1211), (39,6292), (40,6260), (41,1212), (42,1213), (48,1214), (51,233), (54,140), (55,9), (56,1), (57,142), (58,1125), (59,3035), (60,4999), (61,629), (62,630), (64,4), (65,442), (66,427), (67,858), (68,11585), (69,1368), (71,440), (74,30), (75,2887), (76,626), (80,3814), (81,3739), (82,1215), (83,3934), (84,946), (86,3741), (87,3840), (88,3834), (95,3819), (96,1216), (98,511), (99,512), (100,513), (101,514), (102,515), (103,516), (104,517), (105,518), (106,519), (107,520), (108,521), (109,522), (110,523), (111,524), (112,525), (144,2884), (145,2885), (154,1249), (159,3162), (162,8062), (184,216), (186,1511), (187,2482), (190,3835), (197,478), (198,223), (220,6554), (221,7952), (237,11672), (248,441), (249,620), (250,5972), (251,3589), (254,12359), (256,3847), (259,2090), (263,3815), (265,2072), (266,178), (267,11263), (269,11019), (284,5745), (288,6709), (291,3836), (292,3912), (359,1115), (371,641), (372,642), (378,4550), (394,6389), (468,5181), (476,526), (477,5663), (480,5574), (485,639), (486,640), (511,114), (512,115), (513,11), (514,116), (515,117), (516,118), (517,119), (518,120), (519,121), (520,122), (521,123), (522,124), (523,125), (524,126), (525,127), (526,3258), (577,6509), (595,4075), (604,3752), (608,3772), (649,1086), (651,4885), (660,3837), (661,8287), (662,4369), (663,1146), (667,1015), (669,1084), (671,625), (674,5513), (675,674), (681,680), (685,6130), (689,688), (690,5099), (691,690), (692,650), (694,325), (695,6656), (697,696), (699,698), (701,700), (703,702), (705,704), (707,706), (709,708), (711,710), (713,712), (715,714), (717,716), (719,718), (721,720), (723,722), (725,724), (727,726), (729,538), (731,730), (733,732), (735,734), (737,736), (739,536), (741,740), (743,742), (745,744), (747,746), (753,752), (755,754), (759,758), (761,760), (767,766), (769,768), (773,772), (777,776), (779,778), (781,780), (783,782), (785,784), (787,786), (789,788), (791,790), (793,792), (795,794), (797,796), (803,802), (804,2679), (805,804), (807,806), (809,808), (813,812), (814,5509), (815,814), (817,816), (819,818), (825,824), (827,826), (831,830), (833,832), (834,5515), (835,834), (839,838), (840,528), (841,541), (842,542), (843,543), (847,5449), (879,3150), (888,9151), (893,4357), (895,5159), (897,4892), (898,891), (900,3259), (901,900), (902,4370), (904,1107), (907,3800), (909,3911), (911,241), (915,912), (917,916), (919,918), (924,136), (925,924), (926,1566), (927,926), (929,928), (930,1510), (931,8672), (932,4083), (933,6368), (934,3900), (935,9517), (937,9843), (939,936), (940,10472), (941,5743), (943,5044), (945,355), (947,6684), (953,952), (957,495), (961,3812), (963,40), (972,971), (994,3822), (1000,3820), (1001,3789), (1002,3826), (1014,3742), (1015,6547), (1018,4129), (1033,3343), (1035,3341), (1036,958), (1037,1376), (1042,1834), (1057,9708), (1059,9709), (1105,5907), (1113,2574), (1114,2575), (1126,3634), (1138,10264), (1141,1154), (1154,128), (1155,10427), (1156,5087), (1157,6592), (1166,6689), (1167,6700), (1168,6702), (1169,6703), (1170,6706), (1171,6707), (1172,6708), (1173,3628), (1174,6666), (1175,6675), (1176,6676), (1177,468), (1178,6682), (1179,5462), (1220,3831), (1243,6881), (1245,4205), (1252,4422), (1280,3823), (1289,8673), (1290,8674), (1292,3309), (1293,3667), (1294,6000), (1295,6001), (1296,1499), (1297,1503), (1301,8057), (1302,8675), (1304,9033), (1305,8676), (1308,3887), (1309,8677), (1310,8678), (1311,8679), (1318,3036), (1319,1145), (1320,5123), (1333,3666), (1350,7710), (1379,3413), (1380,3414), (1381,3307), (1382,3308), (1383,597), (1384,11165), (1389,9956), (1390,3844), (1402,2092), (1407,4000), (1411,1737), (1412,3946), (1413,3086), (1415,905), (1420,12640), (1436,57), (1438,3008), (1459,2968), (1461,7658), (1464,6739), (1476,5836), (1477,5853), (1485,184), (1486,5452), (1492,4874), (1495,3163), (1498,6523), (1499,5512), (1500,6537), (1501,8265), (1503,132), (1510,137), (1576,647), (1604,6609), (1609,6503), (1611,6338), (1613,6374), (1614,6663), (1616,6552), (1617,6600), (1627,6665), (1634,3005), (1676,5404), (1677,5403), (1691,5976), (1799,11574), (1911,1575), (1916,5031), (1919,6377), (1974,1196), (1976,230), (1979,9296), (1987,297), (1988,264), (1989,3580), (1995,8542), (2053,3061), (2054,10026), (2065,6036), (2070,11597), (2071,11598), (2076,8290), (2078,6594), (2109,335), (2155,1427), (2160,5249), (2161,908), (2162,75), (2163,551), (2164,63), (2165,343), (2176,6376), (2178,6505), (2191,4847), (2192,281), (2207,3767), (2208,1108), (2213,443), (2217,65), (2218,72), (2221,4657), (2222,3738), (2223,6184), (2248,86), (2249,8680), (2258,5257), (2283,3126), (2291,527), (2334,1698), (2346,3740), (2351,577), (2353,32), (2357,1901), (2365,2385), (2366,2386), (2367,2387), (2368,2388), (2369,2389), (2370,2390), (2371,2391), (2372,2392), (2373,2393), (2374,8681), (2375,8682), (2378,530), (2379,531), (2380,532), (2381,533), (2382,537), (2383,539), (2384,545), (2393,1560), (2420,5664), (2489,6388), (2574,1313), (2575,1312), (2605,6741), (2623,338), (2687,2771), (2688,2772), (2689,2773), (2690,2774), (2691,2775), (2692,2776), (2693,2777), (2694,2778), (2695,2779), (2696,2780), (2697,2781), (2698,2782), (2699,2783), (2700,2784), (2701,2785), (2702,2786), (2703,2787), (2704,2788), (2705,2789), (2706,2790), (2707,2791), (2708,2792), (2709,2793), (2710,2794), (2711,2795), (2712,2796), (2713,2797), (2714,2798), (2715,2799), (2716,2800), (2717,2801), (2718,2802), (2719,2803), (2720,2804), (2721,2805), (2722,2806), (2723,2807), (2724,2808), (2725,2809), (2726,2810), (2727,2811), (2728,2812), (2729,2813), (2730,2814), (2731,2815), (2732,2816), (2733,2817), (2734,2818), (2735,2819), (2736,2820), (2737,2821), (2738,2822), (2739,2823), (2740,2824), (2741,2825), (2742,2826), (2743,2827), (2744,2828), (2745,2829), (2746,2830), (2747,2831), (2748,2832), (2749,2833), (2750,2834), (2751,2835), (2752,2836), (2753,2837), (2754,2838), (2755,2839), (2756,2840), (2757,2841), (2758,2842), (2759,2843), (2760,2844), (2761,2845), (2762,2846), (2763,2847), (2764,2848), (2765,2849), (2766,2850), (2767,2851), (2768,2852), (2769,2853), (2770,2854), (2855,2869), (2856,2870), (2857,2871), (2858,2872), (2859,2873), (2860,2874), (2861,2875), (2862,2876), (2863,2877), (2864,2878), (2865,2879), (2866,2880), (2867,2881), (2868,2882), (2980,51), (3052,3161), (3053,6337), (3062,3817), (3065,11813), (3131,10640), (3132,10639), (3155,10962), (3156,10960), (3207,3160), (3222,3221), (3224,76), (3257,4928), (3286,8299), (3304,11530), (3309,5511), (3333,20023), (3345,6245), (3346,6247), (3413,2039), (3414,2040), (3415,1386), (3417,1385), (3420,999), (3422,993), (3423,1001), (3424,5480), (3425,182), (3426,381), (3427,7680), (3429,7683), (3431,549), (3432,54), (3433,55), (3435,56), (3437,58), (3438,15), (3439,16), (3440,13), (3441,14), (3442,61), (3443,62), (3444,81), (3445,8), (3446,100), (3447,110), (3449,6690), (3450,6691), (3451,6692), (3453,6693), (3455,187), (3456,5007), (3457,396), (3458,395), (3489,17), (3490,18), (3518,1493), (3521,10024), (3527,1656), (3531,5055), (3532,20), (3563,3564), (3565,3566), (3566,5139), (3577,10175), (3657,3139), (3667,5510), (3669,4904), (3733,244), (3900,5514), (3939,4521), (4017,8286), (4057,8054), (4076,3038), (4079,6627), (4083,5518), (4557,661), (4559,1577), (4588,4777), (4599,8060), (4606,4940), (4845,5199), (5162,10291), (5210,11147), (5413,8968), (5467,1649), (5504,10257), (5545,4843), (5558,9710), (5562,10600), (5606,8702), (5879,3357), (5961,12095), (5966,5965), (5969,9152), (5970,5969), (5989,9467), (6000,133), (6010,6002), (6011,6003), (6012,6004), (6013,6005), (6014,6006), (6015,6007), (6016,6008), (6017,6009), (6065,3039), (6078,6084), (6079,6085), (6080,6086), (6081,6087), (6082,6088), (6083,6089), (6084,5519), (6085,5516), (6129,7358), (6135,6364), (6136,6365), (6145,1594), (6183,6182), (6186,1100), (6187,44), (6233,8704), (6323,3849), (6325,8705), (6380,6379), (6401,12603), (6402,12604), (6551,6550), (6571,8710), (6573,8711), (6574,8712), (6575,8713), (6577,8714), (6578,6367), (7023,5573), (7037,282), (7050,2345), (7087,31), (7152,278), (7169,19), (7236,750), (7246,748), (7252,4858), (7320,9711), (7953,7927), (7954,7950), (8059,8058), (8064,8063), (8301,9470), (8431,10745), (8576,615), (8577,590), (8615,1104), (8652,4802), (8674,5520), (8676,5190), (8677,10017), (8678,5517), (8684,3808), (8685,3810), (8686,3880), (8687,3910), (8690,4139), (8691,4160), (8693,4762), (8694,4778), (8695,4844), (8696,4912), (8697,4926), (8698,4943), (8699,4962), (8700,4971), (8701,4977), (8702,5952), (8704,12494), (8706,6363), (8707,6371), (8708,6372), (8709,6373), (8750,3239), (8761,92), (8769,4138), (8770,69), (8852,238), (8884,389), (8946,486), (8948,485), (9023,9193), (9056,8999), (9057,9000), (9058,9001), (9059,9002), (9060,9003), (9061,9004), (9062,9005), (9063,9006), (9064,9007), (9065,9008), (9066,9009), (9067,9010), (9068,9011), (9069,9012), (9070,9013), (9071,9014), (9072,9015), (9073,9016), (9074,9017), (9075,9018), (9076,9019), (9077,9020), (9078,9021), (9079,9022), (9080,9023), (9081,9024), (9082,9025), (9083,9026), (9084,9027), (9085,9028), (9086,9029), (9087,9030), (9088,9031), (9089,9032), (9090,9034), (9091,9035), (9092,9036), (9093,9037), (9094,9038), (9096,9040), (9097,9041), (9098,9042), (9099,9043), (9100,9044), (9101,9045), (9102,9046), (9103,9047), (9104,9048), (9105,9049), (9106,9050), (9107,9051), (9108,9052), (9109,9053), (9110,9054), (9111,9055), (9132,9134), (9150,888), (9178,1648), (9192,9189), (9217,99), (9259,6631), (9265,668), (9282,11814), (9292,5254), (9309,3816), (9315,3663), (9412,9410), (9431,9428), (9432,5121), (9468,3229), (9500,673), (9513,11007), (9515,7820), (9831,9830), (10159,7849), (10293,11799), (10307,7956), (10308,9955), (10309,7681), (10415,6698), (10419,6699), (10428,1387), (10511,8262), (10603,8263), (10623,3579), (10630,5461), (11051,7), (11058,7703), (11141,11126), (11142,11127), (11166,11168), (11181,11179), (11270,550), (11589,3184), (11593,11594), (11606,5103), (11636,3906), (11738,3845), (11744,403), (11815,12006), (11816,143), (12028,12358), (12074,12073)
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου