HYACINTHOS 29171

[César Lozada]: 

 

Q1( X(7) ) = MIDPOINT OF X(7) AND X(2886)

= 2*a^5-2*(b+c)*a^4-(3*b^2-4*b*c+3*c^2)*a^3+(b+c)*(b^2-6*b*c+c^2)*a^2+(5*b^2+4*b*c+5*c^2)*(b-c)^2*a-3*(b^2-c^2)*(b-c)^3 : : (barys)

= X(144)-5*X(31245), 3*X(6173)-X(8255)

= lies on these lines: {1, 528}, {7, 2886}, {142, 6690}, {144, 31245}, {527, 3838}, {1376, 30275}, {2550, 5855}, {5173, 15587}, {5735, 28628}, {6667, 8257}, {20008, 30340}

= midpoint of X(i) and X(j) for these {i,j}: {7, 2886}, {5173, 15587}

= reflection of X(6690) in X(142)

= [ 1.6852397298353630, 1.0938821073072020, 2.1055585323092260 ]

 

Q1( X(8) ) = MIDPOINT OF X(8) AND X(1329)

= (2*a^3-2*(b+c)*a^2-(b^2-8*b*c+c^2)*a+3*(b^2-c^2)*(b-c))*(-a+b+c) : : (barys)

= 3*X(8)+X(2098), 3*X(10)-X(24928), X(46)-9*X(3679), X(46)-3*X(8256), X(56)-5*X(3617), X(145)-5*X(31246), 3*X(1329)-X(2098), X(3436)+7*X(4678), 3*X(3679)-X(8256), 3*X(4669)+X(21616), 3*X(5790)-X(7681), 3*X(6691)-2*X(24928)

= lies on these lines: {8, 11}, {10, 6691}, {46, 529}, {56, 3617}, {145, 31246}, {496, 32426}, {517, 546}, {1145, 11010}, {2646, 6735}, {2829, 5690}, {3035, 21842}, {3333, 32049}, {3436, 4678}, {3625, 11545}, {3812, 11035}, {4669, 21616}, {5046, 13996}, {5790, 7681}, {6667, 22837}, {9711, 12647}, {11009, 17757}, {17768, 24393}

= midpoint of X(8) and X(1329)

= reflection of X(6691) in X(10)

= [ 4.1789915178458030, 2.1414732653637700, 0.2293407514960153 ]

 

Q1( X(13) ) = MIDPOINT OF X(13) AND X(623)

= 2*a^6-8*(b^2+c^2)*a^4+(b^4-24*b^2*c^2+c^4)*a^2+5*(b^4-c^4)*(b^2-c^2)-2*sqrt(3)*S*(2*a^4+4*(b^2+c^2)*a^2-3*b^4+8*b^2*c^2-3*c^4) : : (barys)

= 5*S^2-sqrt(3)*(SA-3*SW)*S+9*SB*SC : : (barys)

= 3*X(13)+X(298), 5*X(13)+3*X(21359), X(298)-3*X(623), 5*X(298)-9*X(21359), X(396)-3*X(5459), X(396)+3*X(31693), X(621)+3*X(16267), 5*X(623)-3*X(21359), 3*X(5470)+X(5978)

= lies on these lines: {2, 10646}, {3, 6673}, {5, 6670}, {13, 298}, {14, 7797}, {17, 617}, {30, 6669}, {115, 396}, {511, 20252}, {530, 625}, {533, 11542}, {546, 6694}, {547, 6672}, {618, 5318}, {619, 11303}, {621, 16267}, {629, 16965}, {633, 33465}, {3589, 5066}, {3642, 18582}, {3643, 11305}, {5470, 5978}, {11300, 16966}, {16241, 31704}, {16644, 22917}, {16645, 18584}, {20253, 25555}, {20377, 20415}

= midpoint of X(i) and X(j) for these {i,j}: {13, 623}, {618, 5318}, {619, 31709}, {5459, 31693}

= reflection of X(6671) in X(6669)

= [ 1.2343881651051460, 0.5790897720713462, 2.6700385635018410 ]

 

Q1( X(14) ) = MIDPOINT OF X(13) AND X(624)

= 2*a^6-8*(b^2+c^2)*a^4+(b^4-24*b^2*c^2+c^4)*a^2+5*(b^4-c^4)*(b^2-c^2)+2*sqrt(3)*S*(2*a^4+4*(b^2+c^2)*a^2-3*b^4+8*b^2*c^2-3*c^4) : : (barys)

= 5*S^2+sqrt(3)*(SA-3*SW)*S+9*SB*SC : : (barys)

= 3*X(14)+X(299), 5*X(14)+3*X(21360), X(299)-3*X(624), 5*X(299)-9*X(21360), X(395)-3*X(5460), X(395)+3*X(31694), X(622)+3*X(16268), 5*X(624)-3*X(21360), 3*X(5469)+X(5979)

= lies on these lines: {2, 10645}, {3, 6674}, {5, 6669}, {13, 7797}, {14, 299}, {18, 616}, {30, 6670}, {115, 395}, {511, 20253}, {531, 625}, {532, 11543}, {546, 6695}, {547, 6671}, {618, 11304}, {619, 5321}, {622, 16268}, {630, 16964}, {634, 33464}, {3589, 5066}, {3642, 11306}, {3643, 18581}, {5469, 5979}, {11299, 16967}, {16242, 31703}, {16644, 18584}, {16645, 22872}, {20252, 25555}, {20378, 20416}

= midpoint of X(i) and X(j) for these {i,j}: {14, 624}, {618, 31710}, {619, 5321}, {5460, 31694}

= reflection of X(6672) in X(6670)

= [ 5.8547953543329900, 5.7715952359066320, -3.0572685372585160 ]

 

Q2( X(11) ) = MIDPOINT OF X(11) AND X(59)

= 2*a^8-4*(b+c)*a^7-(b^2-14*b*c+c^2)*a^6+2*(b+c)*(3*b^2-8*b*c+3*c^2)*a^5-(3*b^4+3*c^4+(5*b^2-18*b*c+5*c^2)*b*c)*a^4+5*(b^2-c^2)*(b-c)*b*c*a^3+(b^4+c^4-(b+c)^2*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*(2*b^2+b*c+2*c^2)*a+(b^4-c^4)*(b^2-c^2)*(b-c)^2 : : (barys)

= lies on these lines: {11, 59}, {521, 3035}, {1090, 5723}, {6129, 23593}

= midpoint of X(11) and X(59)

= [ 1.3787767175762080, 1.7312085711800470, 1.8057769860553270 ]

 

Q2( X(24) ) = MIDPOINT OF X(24) AND X(68)

= SA*((12*R^2-SA-SW)*S^2+(SB+SC)*(R^2*(2*R^2-7*SA-SW)+SA^2-SB*SC)) : : (barys)

1216}, {468, 12421}, {597, 8254}, {1147, 13292}, {1495, 32145}, {1899, 32048}, {6353, 12420}, {10020, 22663}, {12134, 32539}, {12163, 31725}

= midpoint of X(i) and X(j) for these {i,j}: {24, 68}, {12163, 31725}

= reflection of X(i) in X(j) for these (i,j): (1147, 16238), (11585, 5449)

= [ 1.4806149089802230, 1.8779964255432660, 1.6571523831558620 ]

 

César Lozada