[Antreas P. Hatzipolakis]:
Let ABC be a triangle, P a point and A'B'C' the pedal triangle of P.
Denote:
(Na),(Nb), (Nc) = the NPCs of PBC, PCA, PAB, resp.
A1, B1, C1 = the midpoints of AP, BP, CP, resp.
The parallel through A1 to BC (and B1C1) intersects again (Nb),(Nc) at Ab, Ac, resp.
The parallel through B1 to CA (and C1A1) intersects again (Nc),(Na) at Bc, Ba, resp.
The parallel through C1 to AB (and A1B1) intersects again (Na),(Nb) at Ca, Cb, resp.
Ma, Mb, Mc = the midpoints of AbAc, BcBa, CaCb, resp.
M1, M2, M3 = the midpoints of BcCb, CaAc, AbBa, resp
Mi, Mii, Miii = the midpoints of BaCa, CbAb, AcBc, resp.
Which is the locusof P such that:
1. The perpendicular bisectors of AbAc, BcBa, CaCb are concurrent?
(I lies on the locus. Point of concurrence = the circumcenter of NaNbNc = N of ABC)
2. The perpendicular bisectors BcCb, CaAc, AbBa are concurrent?
(I lies on the locus)
3. The perpendicular bisectors of BaCa, CbAb, AcBc are concurrent?
4. The triangles A'B'C', MaMbMc are orthologic?
5. The triangles A'B'C', M1M2M3 are orthologic?
6. The triangles A'B'C', MiMiiMiii are orthologic?
[César Lozada]:
Assume P=x:y:z (barys). (All following results are expressed in barycentrics coordinates)
1) Locus = The entire plane
Perspector Q1(P) = (SB+SC)^2*y^2*z^2+(x+y+z)*(S^2+SB*SC)*x*y*z-((SA+SB)*SB*y^2+(SA+SC)*SC*z^2)*x^2 : :
Q1(P) = Midpoint(P, X(4)) if P lies on the circumcircle of ABC.
ETC pairs (P,Q1(P)): (1,5), (2,381), (3,20299), (4,3), (6,24206), (7,1001), (8,12114), (30,32417), (57,3820), (254,20422)
2) Locus = {sidelines of ABC} ∪ {Linf}∪{circumcircle}∪{q7: circum-degree-7 through ETCs 1, 4, 6}
q7: ∑ [y*z*((b^2*c^4*(2*a^2-c^2)*y-b^4*c^2*(2*a^2-b^2)*z)*x^4-(b^2-c^2)*a^2*b^2*c^2*x^3*y*z-a^6*y^3*z^2*c^2+a^6*y^2*z^3*b^2)] = 0
ETC pairs (P, Q2(P)): (1, 10), (4, 5), (6, 24206)
3) Locus = {circumcircle} ∪ {Jerabek hyperbola}
Q3(P) = Midpoint(P, X(4)) if P lies on the circumcircle of ABC.
ETC pairs (P, Q3(P)): (3,5449), (4,5), (6,24206), (54,1209), (64,20299), (65,9956), (66,20300), (67,20301), (68,20302), (70,20303), (74,125), (2574,2574), (2575,2575), (16835,18488)
4) Locus = {circumcircle} ∪ {Thomson cubic} ∪ {q8: degree-8 through ETCs 249, 1138, 23984, 23985 }
q8: ∑ [(a^6*y^3*z^3-2*(6*S^2+3*SA^2+SB*SC)*a^2*x^2*y^2*z^2+(2*SB*b^4*z^2+2*SC*c^4*y^2+(SA-3*SW)*b^2*c^2*y*z)*x^4)*y*z] = 0
If P ∈ q8 then Ma, Mb, Mc are collinear.
Some orthologic centers:
Q4m( X(1) ) = Ma->Ap = X(3884); Q4a( X(1) ) = Ap->Ma = X(13375)
5) Locus = The entire plane.
Orthologic centers:
Q5m(P ) = M1->Ap = (2*x+y+z)*a^2*y*z+(b^2*z+c^2*y)*x^2 : :
Q5a( P ) = Ap->M1 = (a^4*y^2*z^2+(2*(a^2-c^2)*b^2*z+2*(a^2-b^2)*c^2*y)*x*y*z+(b^4*z^2+c^4*y^2)*x^2)*(((b^2+c^2)*a^2-(b^2-c^2)^2)*a^2*y^2*z^2-2*(-b^2*z^2-c^2*y^2+(2*a^2-b^2-c^2)*y*z)*b^2*c^2*x^2+((a^2-b^2+c^2)*b^2*z-(a^2+b^2-c^2)*c^2*y)*(b^2-c^2)*x*y*z)*x*a^2 : :
ETC pairs (P, Q5m): (1,1), (2,597), (3,5), (4,5), (5,8254), (6,597), (7,8255), (8,8256), (9,8257), (10,8258), (13,396), (14,395), (15,396), (16,395), (17,8259), (18,8260), (20,5894), (21,8261), (23,8262), (25,8263), (31,18805), (32,18806), (35,14526), (36,1737), (40,1158), (54,8254), (55,8255), (56,8256), (57,8257), (58,8258), (61,8259), (62,8260), (64,5894), (65,8261), (67,8262), (69,8263), (75,18805), (76,18806), (79,14526), (80,1737), (84,1158), (399,18285), (484,1749), (1054,1052), (1138,18285), (1157,24385), (1263,24385), (3065,1749), (6212,169), (6213,169), (9282,1052) and maybe more
ETC pairs (P, Q5a): (4,52)
6) Locus = { Linf }∪{circumcircle}∪{Thomson cubic}∪{q4: circum-quartic through X(18771) }
If P ∈ {Linf}∪{q4} then Mi, Mii, Miii are collinear.
Some related centers:
Q1 ( X(5) ) = COMPLEMENT OF X(30484)
= S^4-(R^2*(28*R^2+5*SA-24*SW)-2*SA^2+7*SB*SC+5*SW^2)*S^2+(20*R^4-19*R^2*SW+5*SW^2)*SB*SC : :
= 3*X(547)-2*X(32904), 5*X(3858)-X(15619)
= lies on these lines: {2, 30484}, {3, 24573}, {5, 11701}, {30, 13856}, {54, 137}, {546, 8254}, {547, 32904}, {1510, 12006}, {3858, 15619}, {5501, 25340}, {10285, 25150}
= reflection of X(32536) in X(20414)
= complement of X(30484)
= [ -2.5105243195330220, -5.1611547634173640, 8.3724751579039490 ]
Q1 ( X(20) ) = COMPLEMENT OF X(3183)
= S^4-(R^2*(28*R^2+5*SA-24*SW)-2*SA
= lies on these lines: {2, 3183}, {3, 1661}, {4, 1073}, {5, 6523}, {30, 20329}, {64, 122}, {381, 14059}, {382, 10745}, {1032, 3348}, {1657, 6760}, {1853, 8798}, {5895, 14379}, {5925, 12096}, {6260, 18589}
= midpoint of X(4) and X(3346)
= reflection of X(6523) in X(5)
= complement of X(3183)
= {X(5894), X(31377)}-harmonic conjugate of X(3)
= [ 2.0621926634456670, 0.8656351516134604, 2.0895973784309060 ]
Q3( X(265) ) = COMPLEMENT OF X(12893)
=SA*(8*(4*R^2-SA)*S^2+(SB+SC)*((54*R^2-33*SA-29*SW)*R^2+8*SA^2-8*SB*SC+4*SW^2)) : :
= X(68)-5*X(15081), 3*X(381)+X(12302), 5*X(1656)-X(2931), 3*X(2072)+X(32123), 7*X(3090)+X(12319), X(3448)+3*X(5654), 9*X(5055)-X(12310), X(5504)+3*X(14644), X(7689)-3*X(15061), X(9927)-3*X(14644), X(13293)-3*X(18281), 3*X(14643)+X(15133), 5*X(15027)+X(15083), 9*X(23515)-X(32263)
= lies on these lines: {2, 12893}, {3, 19479}, {4, 12901}, {5, 1511}, {68, 15081}, {110, 7577}, {113, 1594}, {125, 1568}, {381, 12302}, {403, 12295}, {539, 11804}, {541, 23315}, {542, 20300}, {858, 16111}, {1209, 10170}, {1656, 2931}, {2777, 13371}, {3047, 25739}, {3090, 12319}, {3448, 5654}, {3564, 20301}, {3574, 16222}, {5055, 12310}, {5448, 5663}, {5449, 11591}, {5504, 9927}, {6639, 22109}, {6699, 11585}, {7689, 15061}, {7741, 12888}, {7951, 19469}, {9820, 32423}, {10024, 16163}, {10182, 13406}, {10254, 12121}, {10264, 22660}, {10576, 12891}, {10577, 12892}, {10663, 16966}, {10664, 16967}, {10721, 31074}, {10733, 16868}, {13289, 18569}, {13293, 18281}, {14643, 15133}, {14708, 18388}, {14915, 15125}, {14984, 15088}, {15027, 15083}, {18404, 32607}, {23336, 25564}
= midpoint of X(i) and X(j) for these {i,j}: {3, 19479}, {4, 12901}, {5, 23306}, {5504, 9927}, {7687, 15115}, {10264, 22660}, {13289, 18569}
= reflection of X(i) in X(j) for these (i,j): (5449, 20304), (25564, 23336), (32743, 10224)
= complement of X(12893)
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 1568, 7723), (5504, 14644, 9927)
= [ 1.7286008730584900, 0.6873112226754586, 2.3670178478743740 ]
Q3( X(290) ) = X(5)X(39) ∩ X(125)X(33330)
= S^6-(20*R^2*SW-SB*SC-4*SW^2)*S^4-(2*R^2*(3*SA^2-5*SA*SW+3*SW^2)-SW*(2*SA^2-2*SA*SW+SW^2))*SW*S^2-(2*R^2-SW)*SB*SC*SW^3 : :
= 5*X(1656)-X(3511)
= lies on these lines: {5, 39}, {125, 33330}, {290, 7752}, {626, 5449}, {1656, 3511}
= [ 1.6149120356671780, 2.0361027141052110, 1.4857108941420740 ]
Q4m( X(2) ) = X(597); Q4a( X(2) ) = X(2)
Q4m( X(3) ) = X(20299)
Q4a( X(3) ) = COMPLEMENT OF X(15319)
= (S^2+(4*R^2+2*SA-3*SW)*SA)*(3*S^2-SB*SC)*SB*SC : :
= X(4)-5*X(3462)
= lies on the cubic K569 and these lines: {2, 15319}, {4, 54}, {133, 16252}, {548, 3184}, {800, 11062}, {1075, 10182}, {3526, 6709}, {5878, 16253}, {10192, 14363}, {10303, 20213}
= complement of X(15319)
= complementary conjugate of X(32767)
= [ -6.1293900515980810, 8.7804691799778640, 0.3908273811218956 ]
Q4m( X(4) ) = X(5562); Q4a( X(4) ) = X(52)
Q4m( X(6) ) = X(599)
Q4a( X(6) ) = X(6)X(5054) ∩ X(18800)X(32455)
= (8*a^2-b^2-c^2)*(2*a^4-7*(b^2+c^2)*a^2+3*(b^2-c^2)^2) : :
= (9*SA-7*SW)*(6*S^2+(SB+SC)*SW) : : (barys)
= lies on these lines: {6, 5054}, {18800, 32455}
= [ 0.0956744896803660, 1.0009946056363530, 2.9035107596913500 ]
Q5a( X(1) ) = X(55)X(16296) ∩ X(72)X(519)
= a*(-a+b+c)*(a^2+(b+c)*a-2*b*c)*((b^2-4*b*c+c^2)*a-b^2*c-b*c^2+c^3+b^3) : :
= lies on these lines: {55, 16296}, {72, 519}, {145, 6018}, {390, 1682}, {3601, 21214}, {3664, 17609}, {5903, 12109}
= [ -25.3209177855415400, 0.7705110014618954, 14.7938112280683900 ]
Q5a( X(2) ) = X(511)X(26869) ∩ X(1196)X(9177)
= a^2*(a^4+2*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*((b^2+c^2)*a^4-4*b^2*c^2*a^2-(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)) : :
= lies on these lines: {511, 26869}, {1196, 9177}, {7484, 9734}
= [ -0.0326309268205581, 0.4779510157978274, 3.3248357448876790 ]
Q5a( X(3) ) = COMPLEMENT OF X(18848)
= (-a^2+b^2+c^2)*(a^6-3*(b^2-c^2)^2*a^2+2*(b^4-c^4)*(b^2-c^2))*(4*a^8-5*(b^2+c^2)*a^6-(b^4-10*b^2*c^2+c^4)*a^4+(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :
= SA*(16*R^2-SA-3*SW)*(2*S^2+(SB+SC)*(16*R^2-SA-4*SW)) : : (barys)
= lies on these lines: {2, 18848}, {4, 12096}, {20, 6389}, {185, 520}, {216, 31829}, {550, 10600}, {5894, 15526}
= complement of X(18848)
= [ 12.8721207499709900, 11.7131817162077100, -10.4094401292998700 ]
Q5a( X(6) ) = X(6)X(5055) ∩ X(193)X(26613)
= (7*a^2-2*b^2-2*c^2)*(4*a^4-5*(b^2+c^2)*a^2+3*(b^2-c^2)^2) : :
= lies on these lines: {6, 5055}, {193, 26613}, {3054, 15850}, {3629, 18800}, {5107, 6776}
= [ 1.3580170236834170, -0.1283555160168229, 3.1027489051421410 ]
Q6m( X(1) ) = X(3626)
Q6a( X(1) ) = X(5)X(7743) ∩ X(5882)X(30198)
= a^2*((b+c)*a+b^2-5*b*c+c^2)*((b+c)*a^2-8*b*c*a-(b+c)*(b^2-4*b*c+c^2)) : :
= lies on these lines: {5, 7743}, {5882, 30198}
= [ -0.1832973437071463, -1.3842109748918980, 4.6835631616205300 ]
Q6m( X(2) ) = X(599); Q6a( X(2) ) = Not interesting
Q6m( X(3) ) = X(548)
Q6a( X(3) ) = COMPLEMENT OF X(32533)
= a^2*(3*a^4-6*(b^2+c^2)*a^2+3*b^4+4*b^2*c^2+3*c^4)*(3*a^4-(b^2+c^2)*a^2-2*(b^2-c^2)^2) : :
= (SB+SC)*(S^2-5*SA^2)*(S^2-5*SB*SC) : : (barys)
= 5*X(631)+3*X(25712)
= lies on these lines: {2, 32533}, {6, 12038}, {20, 113}, {110, 11270}, {140, 16254}, {141, 3530}, {184, 10937}, {206, 14810}, {548, 2883}, {631, 1209}, {960, 17502}, {1092, 11597}, {1511, 7689}, {1658, 6593}, {3526, 18396}, {3528, 6030}, {5206, 11672}, {5447, 6293}, {6759, 11598}, {7488, 17713}, {8542, 20190}, {8717, 17821}, {15083, 15646}, {15606, 18324}, {22966, 32171}
= complement of X(32533)
= [ 4.3898467364517590, 3.1872155514466650, -0.5919524705333564 ]
Q6m( X(4) ) = X(5); Q6a( X(4) ) = X(52)
Q6m( X(6) ) = X(20582); Q6a( X(6) ) = X(6)
César Lozada
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