[Kadir Altintas]:
Let ABC be a triangle with orthocenter H.
DEF is the orthic triangle of ABC
Define Ka, Kaa = the symmedian points of AHE, AHF, resp.
Define Kb,Kbb,Kc,Kcc cyclically
La = the line line KaKaa. Define Lb, Lc cyclically.
Prove: triangle bounded by La,Lb,Lc is perspective with ABC
Romantics of Geometry, problem 2957
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[Ercole Suppa]
Let X be the perspector between ABC and the triangle bounded by La,Lb,Lc. We have:
X = ISOGONAL CONJUGATE OF X(3917)
= (a^2 + b^2) (a^2 + b^2 - c^2) (a^2 + c^2) (a^2 - b^2 + c^2) : : (barys)
= (4 R^2 SB + 4 R^2 SC + SB SC + 4 R^2 SW - SB SW - SC SW - SW^2)S^2 + 2 SB SC SW^2 : : (barys)
= lies on these lines: {2,8801}, {4,83}, {20,26224}, {25,183}, {53,6531}, {82,225}, {95,237}, {107,9076}, {235,14860}, {242,4222}, {251,393}, {297,16890}, {317,20022}, {458,18092}, {648,1843}, {689,2374}, {827,1300}, {847,10594}, {1093,1598}, {1105,3575}, {1217,7487}, {1596,16264}, {1826,2201}, {2052,16277}, {3089,18855}, {4194,27067}, {4200,27005}, {4232,10130}, {6530,6756}, {7716,9308}, {8793,11547}, {9755,14486}, {10002,10548}, {10301,17983}, {12110,14575}, {12122,22078}, {12173,18848}, {14018,29568}, {15424,26863}, {16230,18010}, {16263,18494}, {18105,23290}
= isogonal conjugate of X(3917)
= isotomic conjugate of X(3933)
= barycentric product of X(i) and X(j) for these {i,j}: {4,83}, {19,3112}, {25,308}, {27,18082}, {29,18097}, {82,92}, {107,4580}, {162,18070}, {251,264}, {275,17500}, {286,18098}, {393,1799}, {419,14970}, {689,2489}, {733,17984}, {827,14618}, {1093,28724}, {1176,2052}, {1897,10566}, {1973,18833}, {2373,21459}, {2501,4577}, {4599,24006}, {6330,21458}, {6331,18105}, {6335,18108}, {6531,20022}, {10547,18027}, {10548,14860}, {16277,17907}
= barycentric quotient of X(i) and X(j) for these {i,j}: {2,3933}, {4,141}, {6,3917}, {19,38}, {25,39}, {27,16887}, {28,16696}, {31,4020}, {32,20775}, {82,63}, {83,69}, {92,1930}, {112,1634}, {141,4175}, {158,20883}, {251,3}, {264,8024}, {278,3665}, {281,3703}, {286,16703}, {308,305}, {393,427}, {419,732}, {427,7794}, {428,6292}, {458,14994}, {468,7813}, {523,2525}, {607,3688}, {608,1401}, {648,4576}, {827,4558}, {1096,17442}, {1176,394}, {1194,22424}, {1474,17187}, {1783,4553}, {1799,3926}, {1824,3954}, {1826,15523}, {1843,8041}, {1897,4568}, {1973,1964}, {1974,3051}, {2052,1235}, {2207,1843}, {2333,21035}, {2489,3005}, {2501,826}, {3112,304}, {4577,4563}, {4580,3265}, {4599,4592}, {4628,1331}, {5007,22078}, {6524,27376}, {6531,20021}, {6591,2530}, {6995,8362}, {7009,16720}, {7649,16892}, {8743,3313}, {8744,9019}, {8747,17171}, {8793,23115}, {8882,16030}, {8948,26347}, {10311,14096}, {10547,577}, {10549,21243}, {10551,22416}, {10566,4025}, {14273,14424}, {14569,27371}, {14618,23285}, {14885,22138}, {16277,14376}, {16890,4121}, {17409,23208}, {17500,343}, {18070,14208}, {18082,306}, {18097,307}, {18098,72}, {18099,4019}, {18101,26932}, {18105,647}, {18108,905}, {19118,3787}, {20022,6393}, {21458,441}, {21459,858}, {22105,14417}, {28724,3964}
= trilinear product of X(i) and X(j) for these {i,j}: {4,82}, {19,83}, {19,83}, {25,3112}, {25,3112}, {27,18098}, {28,18082}, {28,18082}, {92,251}, {112,18070}, {158,1176}, {308,1973}, {308,1973}, {811,18105}, {827,24006}, {1096,1799}, {1096,1799}, {1172,18097}, {1172,18097}, {1783,10566}, {1897,18108}, {1974,18833}, {1974,18833}, {2190,17500}, {2190,17500}, {2216,10550}, {2489,4593}, {2489,4593}, {2501,4599}, {2501,4599}, {3405,6531}, {4580,24019}, {4628,17924}, {4628,17924}, {6520,28724}, {6520,28724}, {7012,18101}, {7012,18101}, {8767,21458}, {8767,21458}
= trilinear quotient of X(i) and X(j) for these {i,j}: {1,3917}, {6,4020}, {19,39}, {25,1964}, {28,17187}, {31,20775}, {33,3688}, {34,1401}, {75,3933}, {158,427}, {162,1634}, {273,3665}, {318,3703}, {393,17442}, {419,2236}, {428,17457}, {811,4576}, {827,4575}, {862,4093}, {1096,1843}, {1824,21035}, {1897,4553}, {1930,4175}, {1969,8024}, {1973,3051}, {2052,20883}, {2190,16030}, {2333,21814}, {2501,8061}, {4577,4592}, {6520,27376}, {6591,21123}, {16600,22077}, {17446,22424}, {17469,22078}, {20964,22367}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,17500,10550}, {1176,17500,83}
= (6-9-13) search numbers [-1.38657487002217207, -1.94549581815332300, 5.62750383448537444]
Best regards
Ercole Suppa
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