Denote:
A', B', C' = the intersections of L and BC, CA, AB, resp.
Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.
A", B", C" = the reflections of O in Ma, Mb, Mc, resp.
The circles (A", A"A), (B", B"B), (C", C"C) are coaxial.
The other one?
If L is the trilinear polar of P(u:v:w) the other point of intersection of circles (A", A"A), (B", B"B), (C", C"C) is:
Q = ((b^2-c^2) u (v-w)+a^2 (-u^2+v w):a^2 v (u-w)+c^2 v (-u+w)+b^2 (-v^2+u w):(a^2-b^2) (u-v) w+c^2 (u v-w^2)).
** P=Q if and only if P=X(107)
** The locus of the other point of intersection of the circles (A", A"A), (B", B"B), (C", C"C) when L turn around the centroid is the quartic:
(-a^4+2 a^2 b^2-b^4+2 a^2 c^2-2 b^2 c^2-c^4) x^3 y+(-2 a^4+4 a^2 b^2-2 b^4+c^4) x^2 y^2+(-a^4+2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) x y^3+(-a^4+2 a^2 b^2-b^4+2 a^2 c^2-2 b^2 c^2-c^4) x^3 z+(-a^4+2 a^2 b^2-b^4+2 a^2 c^2-c^4) x^2 y z+(-a^4+2 a^2 b^2-b^4+2 b^2 c^2-c^4) x y^2 z+(-a^4+2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) y^3 z+(-2 a^4+b^4+4 a^2 c^2-2 c^4) x^2 z^2+(-a^4-b^4+2 a^2 c^2+2 b^2 c^2-c^4) x y z^2+(a^4-2 b^4+4 b^2 c^2-2 c^4) y^2 z^2+(-a^4-2 a^2 b^2-b^4+2 a^2 c^2+2 b^2 c^2-c^4) x z^3+(-a^4-2 a^2 b^2-b^4+2 a^2 c^2+2 b^2 c^2-c^4) y z^3=0.
Figure and more information in
http://amontes.webs.ull.es/otr ashtm/HGT2017.htm#HG141117 (In Spanish, Sorry)
[César Lozada]:
Hi Antreas and Angel,
Please allow me to add a pair of notes about the locus of the 2nd point of intersection Q(P):
Let L be the polar trilinear of P=u:v:w (trilinears), as Angel said.
- If P moves on a line, for example the polar trilinear of P0=u0:v0:w0 (trilinears), then Q(P) moves on a circle C0 through H satisfying:
- if P0 = X(264) then C0 has radius=0 . In this case the given circles are tangent at H.
- ETC centers on this line: 297, 525, 850, 2501, 2592, 2593, 3569, 3580, 4391, 5523, 9979, 10015, 13302, 14316, 14618, 14918
- if P0 lies on the Steiner circumellipse then C0 has radius=infinite, ie, C0 is a line which coincides with the radical axis of the circles.
- ETC centers on the Steiner circumellipse: 99, 190, 290, 648, 664, 666, 668, 670, 671, 886, 889, 892, 903, 1121, 1494, 2479, 2480, 2481, 2966, 3225, 3226, 3227, 3228, 4555, 4562, 4569, 4577, 4586, 4597, 5641, 6189, 6190, 6528, 6540, 6606, 6613, 6635, 6648, 9487, 11117, 11118, 14616, 14727, 14728, 14970, 15164, 15165.
- otherwise, C0 has center O*= u0*(b*v0+c*w0)*(S^2+SB*SC)-SA* a*b*c*v0*w0 : : (trilinears)
- If P moves on the MacBeath circumconic then Q(P) lies on the circumcircle of ABC
- ETC centers on the MacBeath circumconic: 110, 287, 648, 651, 677, 895, 1331, 1332, 1797, 1813, 1814, 1815, 2986, 2987, 2988, 2989, 2990, 2991, 4558, 4563, 8115, 8116, 8759, 9190, 13136, 13138, 14919
ETC pairs (P,Q(P)) (excluding 1a): (1,5011), (3,13509), (4,5523), (6,23), (8,5179), (10,5134), (23,6), (30,6794), (69,858), (75,5057), (76,316), (81,1325), (86,5196), (88,1320), (94,265), (97,3484), (110,112), (111,895), (141,5189), (145,8074), (193,468), (194,5167), (249,14366), (287,98), (312,5176), (321,5080), (323,3), (324,6761), (333,7424), (338,3448), (343,3153), (385,1316), (394,2071), (512,14700), (514,6788), (523,6792), (524,2), (538,6787), (543,9144), (599,10989), (648,107), (651,108), (671,671), (694,1916), (895,111), (918,10773), (1016,11607), (1086,149), (1111,10770), (1252,14887), (1331,101), (1332,100), (1797,106), (1813,109), (1814,105), (1815,103), (1976,11610), (1992,7426), (1993,186), (1994,2070), (2340,3730), (2395,13137), (2396,12833), (2407,7471), (2421,7468), (2799,11005), (2986,1300), (2987,3563), (2989,917), (2990,915), (2996,5203), (3124,148), (3125,10769), (3187,242), (3218,1), (3219,3465), (3266,69), (3448,115), (3506,13195), (3621,5199), (3762,10774), (3935,9), (3978,76), (4358,8), (4359,5180), (4558,110), (4563,99), (4858,10777), (5392,5962), (5468,7472), (5483,5535), (5905,1785), (6392,5140), (6515,403), (6542,10), (6650,11599), (7664,11061), (7665,5095), (7774,5112), (7779,11007), (8024,5207), (9716,187), (11004,7575), (11064,20), (11078,13), (11092,14), (11126,15), (11127,16), (13136,1309), (13485,6328), (13492,14262), (13582,1263), (14262,13492), (14463,7876), (14919,74), (14920,5667), (14999,4226), (15066,7464)
Some not-ETC Q(P):
Q( X(5) ) = polar circle-inverse-of-X(6748)
= a^10-3*(b^2+c^2)*a^8+(2*b^4+b^ 2*c^2+2*c^4)*a^6+(b^6-c^6)*(b^ 2-c^2)*a^2-(b^4-c^4)*(b^2-c^2) ^3 : : (barys)
= 4*X(5523)-3*X(6794) = 3*X(6794)-2*X(13509)
= On lines: {4, 6}, {115, 14157}, {237, 5938}, {1141, 2715}, {1625, 3153}, {1968, 12289}, {1970, 12254}, {7748, 12290}
= reflection of X(13509) in X(5523)
= polar circle-inverse-of-X(6748)
= {X(5523), X(13509)}-Harmonic conjugate of X(6794)
= [ -0.192207488961293, 0.06418619410314, 3.684939034741265 ]
Q( X(20) ) = polar circle-inverse-of-X(393)
= (-a^2+b^2+c^2)*(2*a^8-(b^2+c^ 2)*a^6+3*(b^2-c^2)^2*a^4-3*(b^ 4-c^4)*(b^2-c^2)*a^2-(b^2-c^2) ^4) : : (barys)
= X(5523)-3*X(6794) = 3*X(6794)+X(13509)
= On cubics K591, K809 and lines: {4, 6}, {30, 1562}, {184, 15048}, {185, 5305}, {230, 3269}, {441, 525}, {524, 10718}, {1294, 2715}, {1552, 15291}, {1559, 6529}, {3172, 5878}, {7735, 10605}, {7807, 9289}, {10312, 13568}
= polar circle-inverse-of-X(393)
= midpoint of X(i) and X(j) for these {i,j}: {1562, 8779}, {5523, 13509}
= polar circle-inverse-of-X(393)
= {X(6794), X(13509)}-Harmonic conjugate of X(5523)
= [ 1.908703191149817, 2.59066349432803, 0.966188282226597 ]
Q( X(99) ) = polar circle-inverse-of-X(8754)
= (a^6-(b^2+c^2)*a^4+(3*b^4-5*b^ 2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^ 2-c^2))*(a^2-b^2)*(a^2-c^2) : : (barys)
= 4*X(125)-5*X(14061) = 3*X(249)-2*X(7472) = 2*X(265)-3*X(14639) = 3*X(5182)-4*X(6593) = 4*X(5465)-3*X(9166) = 2*X(9140)-3*X(9166) = X(12317)-3*X(14651)
= on cubic K872 and these lines: {4, 542}, {98, 5663}, {99, 110}, {113, 11005}, {115, 3448}, {125, 14061}, {146, 2794}, {148, 14683}, {249, 3566}, {265, 14639}, {399, 2782}, {543, 9143}, {597, 11638}, {691, 1499}, {1576, 14366}, {1632, 14884}, {2854, 10754}, {2930, 5969}, {5182, 6593}, {5465, 9140}, {5642, 11006}, {5655, 6054}, {7727, 10053}, {7728, 10722}, {9180, 14559}, {10620, 12042}, {12188, 12308}, {12317, 14651}
= midpoint of X(i) and X(j) for these {i,j}: {148, 14683}, {12188, 12308}
= reflection of X(i) in X(j) for these (i,j): (99, 110), (671, 9144), (691, 14999), (3448, 115), (6054, 5655), (9140, 5465), (10620, 12042), (10722, 7728), (10753, 9970), (11005, 113), (11006, 5642)
= antigonal conjugate of X(6792)
= polar circle-inverse-of-X(8754)
= [ -1.623603505444731, -0.28464534247236, 4.587082106132042 ]
Q( X(190) ) = polar circle-inverse-of-X(2969)
= (a^4-(b+c)*a^3+(b^2-b*c+c^2)* a^2+2*(b^2-c^2)*(b-c)*a-(b^2- c^2)^2)*(a-b)*(a-c) : : (barys)
= on cubic K299 and these lines: {4, 145}, {11, 3315}, {80, 3120}, {100, 190}, {101, 4120}, {522, 14513}, {901, 3667}, {7972, 10700}, {10773, 12831}
= antigonal conjugate of X(6788)
= polar circle-inverse-of-X(2969)
= [-2.48377725274494, -0.893441943783153, 5.40555994425164]
Q( X(2991) ) = polar circle-inverse-of-X(120)
= (a^3-c*a^2+(b^2-2*b*c-c^2)*a+ c*(b^2+c^2))*(a^2-b^2+c^2)*(a^ 3-b*a^2-(b^2+2*b*c-c^2)*a+b*( b^2+c^2))*(a^2+b^2-c^2)*a: : (barys)
= on the circumcircle and these lines: {2, 5521}, {4, 120}, {21, 3565}, {25, 100}, {28, 99}, {101, 1973}, {108, 6353}, {109, 1395}, {110, 2203}, {186, 2691}, {242, 927}, {403, 10100}, {468, 1290}, {691, 2074}, {919, 5089}, {925, 4228}, {934, 1398}, {1294, 7425}, {1295, 7427}, {1296, 4227}, {1297, 7423}, {1305, 4223}, {2370, 7459}, {2373, 7458}, {4222, 6012}, {4231, 6011}, {4232, 9058}, {7438, 9070}, {7469, 10420}
= orthoptic circle of Steiner inellipse-inverse-of-X(5521)
= polar circle-inverse-of-X(120)
= [ 3.585772879332403, -1.50935365101718, 3.030629526765933 ]
ETC pairs (P0, O*):
(2,381), (4,1351), (6,3095), (7,1482), (8,5779), (67,2080), (69,3), (75,355), (76,1352), (83,5480), (85,5805), (86,946), (95,5), (99,523), (141,6287), (183,7697), (189,2095), (250,2967), (253,382), (264,4), (290,511), (298,5617), (299,5613), (308,6248), (309,12699), (311,6288), (314,5887), (316,9970), (317,155), (319,3652), (320,6265), (322,6259), (325,6033), (327,3818), (340,110), (491,6290), (492,6289), (520,14941), (523,6321), (524,8724), (648,525), (664,522), (666,918), (668,513), (671,524), (693,10738), (850,265), (886,888), (889,891), (892,690), (903,519), (1031,13111), (1121,527), (2373,11799), (2481,518), (2966,2799), (2996,11898), (2998,13108), (3225,698), (3226,726), (3227,536), (3228,538), (3260,7728), (3261,10739), (3262,10742), (3263,10743), (3264,10744), (3265,10745), (3266,10748), (3267,10749), (4373,12645), (4555,900), (4562,812), (4569,3900), (4577,826), (4590,114), (4597,4777), (4998,119), (6563,13556), (7318,11928), (8047,12331), (8048,10680), (8795,52), (8797,3843), (9141,5655), (9473,12188), (13485,399), (13573,13310), (13574,11258), (13577,10679), (14615,5878)
César Lozada
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