[ Tran Quang Hung]:
AoPS
Let ABC be a triangle with circumcenter O and incircl (I). Circle (K_b) passes through A, C and is tangent to (I). Circle (K_c) passes through A, B and is tangent to (I). Then there is a circle (Oa) which is tangent to both circles (K_b), (K_c), (AOB) and (AOC).
*** Define (Ob) and (Oc) cyclically.
AOa, BOb , COc concur in the point W = (r+2R) I - 4r O.
with first barycentric coordinate: a(3a^3+a^2(b+c)+ a(-3b^2+2bc-3c^2)-(b-c)^2(b+c) )
W is the midpoint of X(1420) and X(5128).
W is the reflection of X(i) in X(j), for these {i, j}: {1, 1420}, {9614, 3086}
W lies on lines X(i)X(j) for these {i, j}: {1, 3}, {2, 4292}, {4, 3911}, {5, 9579}, {7, 3523}, {8, 4311}, {9, 474}, {10, 4293}, {20, 1210}, {21, 3306}, {28, 8056}, {30, 9581}, {34, 7501}, {58, 937}, {63, 404}, {72, 3928}, {78, 3218}, {79, 4679}, {80, 5787}, {84, 1728}, {90, 3062}, {100, 6765}, {104, 12650}, {108, 1753}, {109, 602}, {140, 5219}, {142, 3624}, {169, 5030}, {172, 9593}, {200, 4973}, {223, 580}, {226, 631}, {238, 1777}, {255, 269}, {282, 1741}, {329, 6700}, {376, 950}, {377, 5705}, {380, 2260}, {386, 4303}, {388, 6684}, {405, 5437}, {411, 1445}, {443, 1478}, {452, 9843}, {496, 9580}, {498, 5290}, {499, 1699}, {515, 1788}, {516, 3086}, {549, 4654}, {550, 5722}, {553, 3487}, {579, 610}, {908, 6921}, {910, 5022}, {920, 1768}, {938, 3522}, {944, 4848}, {946, 3474}, {956, 1706}, {962, 5265}, {970, 3784}, {978, 1044}, {997, 12526}, {1054, 1722}, {1103, 1106}, {1124, 9616}, {1125, 4295}, {1158, 7995}, {1323, 14256}, {1376, 4662}, {1394, 1465}, {1407, 7078}, {1436, 2270}, {1439, 14528}, {1453, 3752}, {1490, 1708}, {1571, 2242}, {1702, 6502}, {1703, 2067}, {1724, 3182}, {1732, 2173}, {1737, 4299}, {1795, 3345}, {1836, 5433}, {1837, 15326}, {1838, 7490}, {1876, 3515}, {2066, 9582}, {2096, 6260}, {2178, 2324}, {2362, 9583}, {2948, 10081}, {2951, 10092}, {2975, 9352}, {3085, 4298}, {3146, 5704}, {3158, 3555}, {3476, 11362}, {3485, 10165}, {3488, 3528}, {3525, 5714}, {3530, 6147}, {3583, 4333}, {3585, 6826}, {3633, 12437}, {3634, 10590}, {3646, 3683}, {3651, 10382}, {3731, 7523}, {3868, 4855}, {3901, 11570}, {3929, 5044}, {4031, 10299}, {4190, 6734}, {4224, 5272}, {4257, 7520}, {4294, 11019}, {4305, 6738}, {4313, 10304}, {4316, 6869}, {4317, 9588}, {4325, 6885}, {4338, 6892}, {4355, 13407}, {4853, 8666}, {4866, 7284}, {4915, 5288}, {4999, 5880}, {5084, 6692}, {5223, 6763}, {5226, 10303}, {5229, 10175}, {5248, 10582}, {5249, 6910}, {5250, 5253}, {5252, 5771}, {5270, 5726}, {5281, 11037}, {5298, 11376}, {5432, 10404}, {5436, 5439}, {5440, 11523}, {5442, 7951}, {5445, 5791}, {5506, 9814}, {5531, 12757}, {5541, 10074}, {5586, 11551}, {5587, 7354}, {5657, 10106}, {5687, 6762}, {5715, 6833}, {5720, 6924}, {5735, 6966}, {5768, 10573}, {5836, 11194}, {6261, 7098}, {6361, 12053}, {6824, 7988}, {6876, 10393}, {6911, 7330}, {6985, 7171}, {7082, 7701}, {7293, 11337}, {7580, 9841}, {7741, 8727}, {7972, 9945}, {8583, 12514}, {8703, 12433}, {9654, 11231}, {9655, 9956}, {9778, 10624}, {9860, 10089}, {9904, 10091}, {9942, 15071}, {9946, 11571}, {10069, 13174}, {10391, 10399}, {10461, 13588}, {10483, 10826}, {11246, 11375}, {12119, 12832}, {12408, 13312}, {12699, 15325}, {13117, 13221}.
(6 - 9 - 13) - search numbers of W: (-1.83665556655288, -1.22199792675032, 5.33435023114362)
Figures and details in
http://amontes.webs.ull.es/ otrashtm/HGT2018.htm#HG060118
Angel Montesdeoca
AoPS
Let ABC be a triangle with circumcenter O and incircl (I). Circle (K_b) passes through A, C and is tangent to (I). Circle (K_c) passes through A, B and is tangent to (I). Then there is a circle (Oa) which is tangent to both circles (K_b), (K_c), (AOB) and (AOC).
*** Define (Ob) and (Oc) cyclically.
AOa, BOb , COc concur in the point W = (r+2R) I - 4r O.
with first barycentric coordinate: a(3a^3+a^2(b+c)+ a(-3b^2+2bc-3c^2)-(b-c)^2(b+c) )
W is the midpoint of X(1420) and X(5128).
W is the reflection of X(i) in X(j), for these {i, j}: {1, 1420}, {9614, 3086}
W lies on lines X(i)X(j) for these {i, j}: {1, 3}, {2, 4292}, {4, 3911}, {5, 9579}, {7, 3523}, {8, 4311}, {9, 474}, {10, 4293}, {20, 1210}, {21, 3306}, {28, 8056}, {30, 9581}, {34, 7501}, {58, 937}, {63, 404}, {72, 3928}, {78, 3218}, {79, 4679}, {80, 5787}, {84, 1728}, {90, 3062}, {100, 6765}, {104, 12650}, {108, 1753}, {109, 602}, {140, 5219}, {142, 3624}, {169, 5030}, {172, 9593}, {200, 4973}, {223, 580}, {226, 631}, {238, 1777}, {255, 269}, {282, 1741}, {329, 6700}, {376, 950}, {377, 5705}, {380, 2260}, {386, 4303}, {388, 6684}, {405, 5437}, {411, 1445}, {443, 1478}, {452, 9843}, {496, 9580}, {498, 5290}, {499, 1699}, {515, 1788}, {516, 3086}, {549, 4654}, {550, 5722}, {553, 3487}, {579, 610}, {908, 6921}, {910, 5022}, {920, 1768}, {938, 3522}, {944, 4848}, {946, 3474}, {956, 1706}, {962, 5265}, {970, 3784}, {978, 1044}, {997, 12526}, {1054, 1722}, {1103, 1106}, {1124, 9616}, {1125, 4295}, {1158, 7995}, {1323, 14256}, {1376, 4662}, {1394, 1465}, {1407, 7078}, {1436, 2270}, {1439, 14528}, {1453, 3752}, {1490, 1708}, {1571, 2242}, {1702, 6502}, {1703, 2067}, {1724, 3182}, {1732, 2173}, {1737, 4299}, {1795, 3345}, {1836, 5433}, {1837, 15326}, {1838, 7490}, {1876, 3515}, {2066, 9582}, {2096, 6260}, {2178, 2324}, {2362, 9583}, {2948, 10081}, {2951, 10092}, {2975, 9352}, {3085, 4298}, {3146, 5704}, {3158, 3555}, {3476, 11362}, {3485, 10165}, {3488, 3528}, {3525, 5714}, {3530, 6147}, {3583, 4333}, {3585, 6826}, {3633, 12437}, {3634, 10590}, {3646, 3683}, {3651, 10382}, {3731, 7523}, {3868, 4855}, {3901, 11570}, {3929, 5044}, {4031, 10299}, {4190, 6734}, {4224, 5272}, {4257, 7520}, {4294, 11019}, {4305, 6738}, {4313, 10304}, {4316, 6869}, {4317, 9588}, {4325, 6885}, {4338, 6892}, {4355, 13407}, {4853, 8666}, {4866, 7284}, {4915, 5288}, {4999, 5880}, {5084, 6692}, {5223, 6763}, {5226, 10303}, {5229, 10175}, {5248, 10582}, {5249, 6910}, {5250, 5253}, {5252, 5771}, {5270, 5726}, {5281, 11037}, {5298, 11376}, {5432, 10404}, {5436, 5439}, {5440, 11523}, {5442, 7951}, {5445, 5791}, {5506, 9814}, {5531, 12757}, {5541, 10074}, {5586, 11551}, {5587, 7354}, {5657, 10106}, {5687, 6762}, {5715, 6833}, {5720, 6924}, {5735, 6966}, {5768, 10573}, {5836, 11194}, {6261, 7098}, {6361, 12053}, {6824, 7988}, {6876, 10393}, {6911, 7330}, {6985, 7171}, {7082, 7701}, {7293, 11337}, {7580, 9841}, {7741, 8727}, {7972, 9945}, {8583, 12514}, {8703, 12433}, {9654, 11231}, {9655, 9956}, {9778, 10624}, {9860, 10089}, {9904, 10091}, {9942, 15071}, {9946, 11571}, {10069, 13174}, {10391, 10399}, {10461, 13588}, {10483, 10826}, {11246, 11375}, {12119, 12832}, {12408, 13312}, {12699, 15325}, {13117, 13221}.
(6 - 9 - 13) - search numbers of W: (-1.83665556655288, -1.22199792675032, 5.33435023114362)
Figures and details in
http://amontes.webs.ull.es/ otrashtm/HGT2018.htm#HG060118
Angel Montesdeoca
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