[Kadir Altintas]:
Let ABC be a triangle, P a point and DEF the cevian triangle of P.
Circle wa is tangent to BC at D and internally tangent to circumcircle of ABC.
Define wb,wc cyclically.
Prove: the circle internally tangent to wa,wb,wc is tangent to incircle of ABC at a point X.
Reference: L. A. Emelianov, Journal "Matematicheskoe Prosvechenie" Tret'ya Seriya, N.7, 2003, Problem section, problem 8
Is there any properties of X, ıf P is X(1), X(2),...of ABC. etc. ?
[Francisco Javier Garcia Capitan]
If P = (u:v:w), then the contact point with the incircle is the point X = ((s-a)((s-b)v-(s-c)w)^2 u^2 : :
(Romantics of Geometry #2853)
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[Ercole Suppa]
*** Some pairs {P=X(i), X=X(j)}:
{1,3022},{2,11},{3,15616},{4,3022},{6,15615},{8,3022},{9,3022},{21,3022},{27,11},{56,1356},{57,1357},{59,15615},{60,15616},{63,1364},{65,1356},{69,1364},{75,11},{77,1364},{79,3022},{80,3022},{81,1364},{84,3022},{85,1358},{86,11},{88,5577},{89,3025},{90,3022},{92,3318},{100,5580},{104,3022},{174,10501},{177,3022},{189,1364},{190,3021},{222,1363},{226,1365},{234,11},{253,3318},{256,3022},{264,3326},{269,1357},{272,11},{273,11},{274,3026},{278,1365},{279,1358},{286,1364},{294,3022},{307,1367},{309,3318},{310,11},{314,3022},{319,3024},{320,3025},{322,3318},{329,3318},{330,3023},{331,1367},{334,3020},{335,11},{342,3318},{344,16184},{347,3318},{348,1367},{479,1357},{508,5997},{513,15615},{514,3328},{518,15615},{527,3328},{554,11},{555,10491},{559,3024},{608,1356},{651,1362},{653,1360},{655,3322},{658,3321},{664,1317},{668,6018},{672,15615},{673,11},{675,11},{679,3025},{693,3326},{749,5581},{757,3025},{799,5579},{840,15615},{870,3023},{871,11},{883,1362},{885,3022},{894,3023},{903,11},{908,3326},{927,3322},{934,1361},{941,3022},{942,15616},{943,3022},{959,1356},{969,1364},{977,6020},{981,3022},{983,3022},{987,3022},{989,3022},{1000,3022},{1002,15615},{1014,1357},{1025,1362},{1029,3024},{1037,15615},{1039,3022},{1041,3022},{1042,1356},{1061,3022},{1063,3022},{1081,11},{1082,3024},{1088,11},{1118,1356},{1119,1357},{1143,10501},{1156,3022},{1172,3022},{1223,11},{1231,1367},{1240,11},{1246,11},{1251,3022},{1266,16185},{1268,11},{1274,10501},{1275,3323},{1320,3022},{1323,3328},{1389,3022},{1392,3022},{1396,1357},{1400,1356},{1414,3028},{1426,1356},{1434,1358},{1439,1363},{1440,11},{1441,1365},{1442,3024},{1443,3025},{1444,1364},{1446,1367},{1458,15615},{1462,1357},{1476,3022},{1488,10501},{1489,10501},{1659,11},{1804,1363},{1814,1364},{1821,5578},{1847,1358},{1876,15615},{1896,3022},{1909,3023},{1937,3022},{2091,1357},{2113,15615},{2296,11},{2298,3022},{2320,3022},{2335,3022},{2344,3022},{2346,3022},{2358,1356},{2363,6020},{2369,1358},{2400,11},{2406,1359},{2481,3022},{2648,3022},{2861,3326},{2989,11},{2994,3024},{2995,1364},{2997,3022},{3062,3022},{3065,3022},{3218,3025},{3219,3024},{3252,15615},{3254,3022},{3255,3022},{3262,3326},{3286,15615},{3296,3022},{3306,5577},{3307,3022},{3308,3022},{3423,15615},{3427,3022},{3433,15615},{3446,15615},{3449,15615},{3467,3022},{3495,3022},{3513,15615},{3514,15615},{3551,3022},{3577,3022},{3662,3020},{3668,1365},{3676,14027},{3680,3022},{3911,14027},{3952,6019},{4146,10504},{4180,3022},{4296,6020},{4303,15616},{4373,11},{4552,3027},{4565,1355},{4566,3028},{4569,1362},{4573,1366},{4618,6018},{4625,3027},{4626,1360},{4866,3022},{4876,3022},{4900,3022},{4998,14027},{5279,6020},{5377,3022},{5424,3022},{5551,3022},{5553,3022},{5555,3022},{5556,3022},{5557,3022},{5558,3022},{5559,3022},{5560,3022},{5561,3022},{5665,3022},{5936,11},{6063,1365},{6183,1362},{6384,11},{6548,11},{6595,3022},{6596,3022},{6597,3022},{6598,3022},{6599,3022},{6601,3022},{6625,3023},{6650,11},{7002,10501},{7003,3022},{7029,3023},{7030,3023},{7045,3025},{7048,10501},{7049,3022},{7055,1363},{7056,1364},{7091,3022},{7126,3022},{7133,3022},{7149,3022},{7155,3022},{7160,3022},{7161,3022},{7162,3022},{7176,3023},{7185,3020},{7192,3025},{7209,1358},{7219,6020},{7224,1364},{7233,1358},{7249,11},{7261,3022},{7270,6020},{7282,3024},{7284,3022},{7285,3022},{7316,1356},{7317,3022},{7318,11},{7319,3022},{7320,3022},{7331,3024},{7357,3024},{7361,7158},{7371,12809},{7595,3022},{7707,3022},{8044,3024},{8046,3025},{8047,3025},{8048,1364},{8049,11},{8050,6021},{8051,1357},{8269,1362},{8372,3022},{8759,3022},{8809,3022},{8810,1363},{8814,1357},{8817,1357},{8822,3318},{8829,1357},{9309,15615},{9343,15615},{9365,3022},{9372,3022},{9436,3323},{9442,3022},{9500,15615},{10029,3323},{10266,3022},{10305,3022},{10307,3022},{10308,3022},{10309,3022},{10390,3022},{10429,3022},{10435,3022},{10436,3026},{10509,1358},{11279,3022},{11604,3022},{11609,3022},{12641,3022},{12867,3022},{12868,3022},{13143,3022},{13386,1364},{13387,1364},{13388,1364},{13389,1364},{13390,11},{13426,3022},{13437,1357},{13454,3022},{13459,1357},{13476,15615},{13577,1364},{13602,3022},{13606,3022},{14224,3022},{14496,3022},{14497,3022},{14616,3024},{14621,11},{14626,15615},{14947,3022},{15173,3022},{15175,3022},{15176,3022},{15179,3022},{15180,3022},{15314,3022},{15315,3022},{15382,15615},{15446,3022},{15467,11},{15728,1357},{15909,3022},{15910,3022},{15997,3022},{15998,3022},{16005,3022},{16078,11},{16099,11},{16615,3022},{17097,3022},{17098,3022},{17103,3023},{17139,3326},{17501,3022},{18026,1361},{18299,3022},{18490,3022},{18771,15615},{18810,3328},{18811,14027},{18815,11},{18816,1364},{18821,3328},{18827,3023},{18884,11},{18886,10504},{19296,12809},{19551,3022},{19604,1357},{19975,11},{20028,11},{20527,11},{20615,1356},{21398,3022},{21446,1357},{21453,11},{21456,10491},{22464,3326},{23062,1358},{23618,1358},{23836,3022},{23838,3022},{23893,3022},{23959,3022},{23973,1360},{24002,3323},{24029,1361},{24149,3327},{24154,11},{24155,11},{24297,3022},{24298,3022},{24300,3022},{24302,3022},{25417,3024},{26722,3022},{26749,3025},{26751,3024},{27447,11},{27475,11},{27483,11},{27494,11},{27498,11},{27818,1358},{28626,11},{28630,3023}
*** Some points:
-- if P = X(5) = The nine point circle center, then
X(P) = (name pending)
= (a-b-c)^3 (b-c)^2 (2 a^3+a^2 b-2 a b^2-b^3+a^2 c+2 a b c+b^2 c-2 a c^2+b c^2-c^3)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4)^2 : : (barys)
= lies on the incircle
= (6-9-13) search numbers [0.647789655574663205, 0.0222078448030230859, 3.32630997908628780]
-- if P = X(7) = Gergonne point the circle internally tangent to wa,wb,wc is the incircle, so point X is undefined
-- if P = X(10) = Spieker center, then
X(P) = X(11)X(24285) ∩ X(55)X(15322)
= (a - b - c) (b - c)^2 (b + c)^2 (3 a + b + c)^2 : : (barys)
= lies on the incircle and these lines: {11,24185}, {55,15322}, {1358,4934}
= barycentric product of X(i) and X(j) for these {i,j}: {4841, 4843}
= barycentric quotient of X(i) and X(j) for these {i,j}: {4832, 5545}, {4843, 4633}, {8653, 4627}
= (6-9-13) search numbers [2.67609161326487573, 3.36839763383821650, 0.0735776831279788791]
-- if P = X(11) = Feuerbach point, then
X(P) = (name pending)
= (a-b-c)^5 (b-c)^6 (2 a^2-a b-b^2-a c+2 b c-c^2)^2 : : (barys)
= lies on the incircle and this line: {3022,11193}
= (6-9-13) search numbers [3.03929782441394446, 3.23250482308499077, 0.0000236854267957403879]
-- if P = X(19) = Clawson point, then
X(P)= X(55)X(28847) ∩ X(56)X(28848)
= a^2 (a-b-c) (b-c)^2 (a^2-b^2-c^2)^2 (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 (a^3-a^2 b+a b^2-b^3-a^2 c+2 a b c+b^2 c+a c^2+b c^2-c^3)^2 : : (barys)
= lies on the incircle and these lines: {55,28847}, {56,28848}, {1362,6180}, {3271,15615}, {3323,4014}
= (6-9-13) search numbers [1.96393129616996292, 3.28399484872213694, 0.460699757636762947]
-- if P = X(20) = DeLongchamps point, then
X(P) = X(1360)X(6284) ∩ X(1362)X(12680)
= 4 (a-b-c)^3 (b-c)^2 (a^3-a^2 b-a b^2+b^3-a^2 c-2 a b c-b^2 c-a c^2-b c^2+c^3)^2 (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4)^2 : : (barys)
= lies on the incircle and these lines: {1360,6284}, {1362,12680}, {3022,4081}
= barycentric product of X(i) and X(j) for these {i,j}: {1097,3119}, {1146,6060}, {7338,23970}
= barycentric quotient of X(i) and X(j) for these {i,j}: {6060,1275}, {7338,23586}
= trilinear product of X(i) and X(j) for these {i,j}: {1097,3022},{1097,3022}, {2310,6060} ,{2310,6060}, {7338,24010}, {7338,24010}
= trilinear quotient of X(i) and X(j) for these {i,j}: {6060,7045}, {7338,24013}
= (6-9-13) search numbers [3.11821797687729470, 3.17062223857958783, 0.00644078894745151748]
Best regards
Ercole Suppa
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