Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
Oa, Ob, Oc = the circumcenters of PB'C', PC'A', PA'B', resp. ( = midpoints of AP, BP, CP, resp.)
Ooa, Oob, Ooc = the circumcenters of PObOc, POcOa, POaOb, resp.
Which is the locus of P such that ..... are perspective?
1. OoaOobOoc, ABC
O, H, I lie on the locus
2. OoaOobOoc, OaObOc.
O, I lie on the locus
3. OoaOobOoc, A'B'C'
O, I, N lie on the locus
[César Lozada]:
1) Locus = uncatalogued cubic pK(X6,X382) = CyclicSum[ (S^2-5*SB*SC)*x*(c^2*y^2-b^2* z^2) = 0 (barycentrics), through ETC’s 1, 3, 4, 382.
ETC pairs (P,Z1(P)): (1,1), (3,3431), (4,381), (382,4)
2) Locus = Neuberg cubics. ETC pairs (P,Z2(P)): (1,1), (4,546)
Z2(O) = Midpoint of {X(3), X(54)}
= (cos(2*A)+3/2)*cos(B-C)-cos(3* A) : : (trilinears)
= On lines: {2,6288}, {3,54}, {4,7712}, {5,5944}, {30,3574}, {35,20014}, {55,10082}, {56,10066}, {125,128}, {143,567}, {156,7503}, {182,9977}, {184,5876}, {186,6152}, {511,10115}, {539,549}, {569,973}, {578,7502}, {631,2888}, {1199,3581}, {1495,3850}, {1498,7526}, {1539,5893}, {2070,10095}, {2917,6644}, {3357,10274}, {3431,3519}, {3576,9905}, {3630,5092}, {5010,6286}, {5462,7575}, {5544,6642}, {5609,5907}, {5888,7666}, {5892,6153}, {7280,7356}, {7583,8995}, {7979,10246}, {10024,10113}
= midpoint of X(i),X(j) for these {i,j}: {3,54}, {195,7691}
= reflection of X(i) in X(j) for these (i,j): (5,6689), (1209,140), (1493,54), (3574,8254)
= Complement of X(6288)
= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (3,195,7691), (54,7691,195), (567,7488,143), (569,1658,5946), (578,7502,10263)
= [ 0.297301920863813, 8.32466091302025, -2.259778575197556 ]
Z2( X(13) ) = Midpoint of {X(13), X(17)}
= -sqrt(3)*((-3*cos(2*A)+5)*cos( B-C)+cos(A)*cos(2*(B-C))+2* cos(A)-cos(3*A))-9*sin(A)+sin( 3*A)-4*sin(A)*cos(2*(B-C))-5* sin(2*A)*cos(B-C) : : (trilinears)
= On lines: {3,13}, {115,6783}, {396,5478}, {397,629}, {532,5459}, {618,6673}, {3054,6115}, {5472,5617}, {6770,7694}
= midpoint of X(i),X(j) for this {i,j}: {13,17}
= reflection of X(i) in X(j) for these (i,j): (618,6673), (629,6669)
= [ 0.731299850937217, 0.64475403461158, 2.856773295974568 ]
Z2( X(14) ) = Midpoint of {X(14), X(18)}
= +sqrt(3)*((-3*cos(2*A)+5)*cos( B-C)+cos(A)*cos(2*(B-C))+2* cos(A)-cos(3*A))-9*sin(A)+sin( 3*A)-4*sin(A)*cos(2*(B-C))-5* sin(2*A)*cos(B-C) : : (trilinears)
= On lines: {3,14}, {115,6782}, {395,5479}, {398,630}, {533,5460}, {619,6674}, {3054,6114}, {5471,5613}, {6773,7694}
= midpoint of X(i),X(j) for this {i,j}: {14,18}
= reflection of X(i) in X(j) for these (i,j): (619,6674), (630,6670)
= [ -5.953621112882535, 10.14824254625153, -0.637216767243975 ]
Z2( X(15) ) = Midpoint of {X(15), X(61)}
= (a^4-(b^2+c^2)*a^2-2*b^2*c^2- sqrt(3)*S*(b^2+c^2))*a : : (trilinears)
= On lines: {3,6}, {23,2981}, {114,6109}, {396,7684}, {398,623}, {621,7834}, {628,7849}, {633,3788}, {635,6671}
= midpoint of X(i),X(j) for this {i,j}: {15,61}
= reflection of X(i) in X(j) for these (i,j): (623,6694), (635,6671)
= [ 2.512076150757173, 2.64540036069190, 0.649813701078983 ]
Z2( X(16) ) = Midpoint of {X(16), X(62)}
= (a^4-(b^2+c^2)*a^2-2*b^2*c^2+ sqrt(3)*S*(b^2+c^2))*a : : (trilinears)
= On lines: {3,6}, {23,6151}, {114,6108}, {395,7685}, {397,624}, {622,7834}, {627,7849}, {634,3788}, {636,6672}
= midpoint of X(i),X(j) for this {i,j}: {16,62}
= reflection of X(i) in X(j) for these (i,j): (624,6695), (636,6672)
= [ -3.579456503575973, -1.99005963865096, 6.670454925700879 ]
3) Locus = Napoleon-Feuerbach cubic. ETC pairs (P, Z3(P)): (1,1319), (3,3), (4,113), (54,30)
Z3( X(5) ) = Midpoint of {X(5), X(6150)}
= (2*cos(2*A)+cos(2*(B-C)))*(2* cos(2*A)*cos(B-C)+cos(3*A)) : : (trilinears)
= On lines: {2,1157}, {5,6150}, {128,539}, {136,186}, {140,389}, {252,1209}
= midpoint of X(i),X(j) for this {i,j}: {5,6150}
= [ 3.125121786478150, 2.99167492935359, 0.127141013980441 ]
Z3( X(17) ) = (3*(SA-SW) + 2*sqrt(3)*S)*( 2*SA-SW – sqrt(3)*S) : : (barycentrics)
= On lines: {16,396}, {141,1078}, {187,624}, {230,5981}, {395,533}, {511,8259}, {623,7749}, {5321,6109}
= [ -0.047312781613489, 0.40997832028103, 3.378670005534510 ]
Z3( X(18) ) = (3*(SA-SW) - 2*sqrt(3)*S)*( 2*SA-SW + sqrt(3)*S) : : (barycentrics)
= On lines: {15,395}, {141,1078}, {187,623}, {230,5980}, {396,532}, {511,8260}, {624,7749}, {5318,6108}
= [ -38.140632978626350, -15.66383711020623, 32.088228471416230 ]
César Lozada
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