[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.
(Oa), (Ob), (Oc) = the concentric circles (P, PNa), (P, PNb), (P, PNc), resp.
Ra, Rb, Rc = the radical axes of ((Na), (Oa)), ((Nb), (Ob)), ((Nc), (Oc)), resp.
A*B*C* = the triangle bounded by Ra, Rb, Rc, resp.
A", B", C" = the intersections Ra /\ BC, Rb /\ CA, Rc /\ AB, resp.
1. P = O:
ABC, A*B*C* are perspective (homothetic).
1.1. Homothetic center?
A'B'C', A*B*C* are perspective (homothetic).
1.2. Homothetic center?
A'B'C' (and ABC), A*B*C* are orthologic.
Orthologic center (A'B'C', A*B*C*) = orthocenter of A'B'C' = O
1.3. Orthologic center (A*B*C*, A'B'C') [= orthologic center (A*B*C*, ABC)] = orthocenter of A*B*C* = ?
2. P = I:
A", B", C" are collinear.
2.1. Line (point it is the trilinear pole of)
ABC, A*B*C* are perspective.
2.2. Perspector?
A'B'C', A*B*C* are orthologic.
Orthologic center (A'B'C', A*B*C*) = antipode of Feuerbach point in the incircle = X(1317)
2.3. Orthologic center (A*B*C*, A'B'C') ?
A'B'C', A*B*C* are parallelogic
Parallelogic center (A'B'C', A*B*C*) = Feuerbach point X(11)
2.4. Parallelogic center (A*B*C*, A'B'C') ?
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[Ercole Suppa]
(1) P=O
*** 1.1
P1 = Homothetic center (ABC, A*B*C*) = X(4)X(94) ∩ X(93)X(22823)
= (3+4*cos(2*A))*(cos(A)-cos(B-C))*(1+2*cos(2*A)-4*cos(A)*cos(B-C)+2*cos(2*(B-C))) : : (trilinears)
= 2*a^16-6*a^14*(b^2+c^2)-7*a^8*b^2*c^2*(b^4-3*b^2*c^2+c^4)-(b^2-c^2)^6*(2*b^4+3*b^2*c^2+2*c^4)+a^12*(4*b^4+17*b^2*c^2+4*c^4)-2*a^6*(b^2-c^2)^2*(b^6-3*b^4*c^2-3*b^2*c^4+c^6)+a^10*(2*b^6-13*b^4*c^2-13*b^2*c^4+2*c^6)+a^2*(b^2-c^2)^4*(6*b^6+b^4*c^2+b^2*c^4+6*c^6)-a^4*(b^2-c^2)^2*(4*b^8-5*b^6*c^2+9*b^4*c^4-5*b^2*c^6+4*c^8) : : (barys)
= lies on these lines: {4,94},{93,22823},{275,1141},{317,328},{1989,3087},{3520, 5961},{5627,10152},{10733,15469}
={X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,265,6344}
=(6-9-13)-search numbers [0.345580867016579927, 0.435031598698632648, 3.17998989803151961]
*** 1.2
P2 = Homothetic center (A'B'C', A*B*C*)
= cos(A)*(-1+2*cos(A))*(1+2*cos(A))*(-4-6*cos(2*A)-2*cos(4*A)+10*cos(A)*cos(B-C)+4*cos(3*A)*cos(B-C)-cos(2*(B-C))-4*cos(A)*cos(3*(B-C))+2*cos(4*(B-C))) : : (trilinears)
= -2 a^22+12 a^20 (b^2+c^2)-4 a^18 (7 b^4+15 b^2 c^2+7 c^4)-2 (b^2-c^2)^8 (b^6+2 b^4 c^2+2 b^2 c^4+c^6)+5 a^16 (6 b^6+23 b^4 c^2+23 b^2 c^4+6 c^6)+4 a^2 (b^2-c^2)^6 (3 b^8+4 b^6 c^2+4 b^4 c^4+4 b^2 c^6+3 c^8)-2 a^14 (6 b^8+49 b^6 c^2+90 b^4 c^4+49 b^2 c^6+6 c^8)+2 a^10 b^2 c^2 (8 b^8-13 b^6 c^2-43 b^4 c^4-13 b^2 c^6+8 c^8)+a^12 (21 b^8 c^2+128 b^6 c^4+128 b^4 c^6+21 b^2 c^8)-a^4 (b^2-c^2)^4 (28 b^10+9 b^8 c^2+8 b^6 c^4+8 b^4 c^6+9 b^2 c^8+28 c^10)+6 a^6 (b^2-c^2)^2 (5 b^12-5 b^10 c^2+2 b^8 c^4+2 b^6 c^6+2 b^4 c^8-5 b^2 c^10+5 c^12)+a^8 (-12 b^14+25 b^12 c^2-43 b^10 c^4+36 b^8 c^6+36 b^6 c^8-43 b^4 c^10+25 b^2 c^12-12 c^14) : : (barys)
=(6-9-13)-search numbers [4.07619883096680966, 2.58781917392769904, -0.0322251758733246926]
*** 1.3
P3 = Orthologic center (A*B*C*, A'B'C') = X(4)X(94) ∩ X(1511)X(22823)
= 2*cos(5*A)-19*cos(B-C)-26*cos(2*A)*cos(B-C)-8*cos(4*A)*cos(B-C)
+2*cos(3*A)*(5+4*cos(2*(B-C)))+2*cos(A)*(11+8*cos(2*(B-C)))-2*cos(3*(B-C))-4*cos(2*A)*cos(3*(B-C)) : : (trilinears)
= -2 a^16+6 a^14 (b^2+c^2)+2 (b^2-c^2)^6 (b^4+b^2 c^2+c^4)-4 a^12 (b^4+4 b^2 c^2+c^4)+a^10 (-2 b^6+11 b^4 c^2+11 b^2 c^4-2 c^6)-6 a^2 (b^2-c^2)^4 (b^6+c^6)+a^8 (9 b^6 c^2-20 b^4 c^4+9 b^2 c^6)+a^4 (b^2-c^2)^2 (4 b^8-3 b^6 c^2+9 b^4 c^4-3 b^2 c^6+4 c^8)+a^6 (2 b^10-13 b^8 c^2+10 b^6 c^4+10 b^4 c^6-13 b^2 c^8+2 c^10) : : (barys)
= lies on these lines: {4,94},{1511,22823}
=(6-9-13)-search numbers [37.1836763346179156, 43.0475945420524932, -43.3232131247217695]
-----------------------
(2)
*** 2.1
A", B", C" are collinear. The line A''B''C'' is the trilinear polar of the point
P4 = X(1)X(4744) ∩ X(2)X(7)
= -3+4*cos(A) : : (barys)
=a (-2 a^2+2 b^2-3 b c+2 c^2),b (2 a^2-2 b^2-3 a c+2 c^2),c (2 a^2-3 a b+2 b^2-2 c^2) : : (barys)
= lies on these lines: {1, 4744}, {2, 7}, {8, 3336}, {21, 5708}, {23, 1473}, {31, 18201}, {40, 3623}, {46, 145}, {56, 18419}, {72, 17572}, {75, 5372}, {81, 89}, {84, 17578}, {88, 4383}, {100, 4430}, {149, 3474}, {165, 3957}, {171, 4392}, {189, 21739}, {191, 5550}, {222, 1994}, {238, 9335}, {244, 4650}, {404, 3940}, {484, 3241}, {499, 1749}, {518, 9352}, {750, 7226}, {896, 17063}, {938, 15680}, {942, 4189}, {982, 17024}, {1155, 3873}, {1376, 4661}, {1407, 1993}, {1454, 3600}, {1621, 4860}, {1707, 7292}, {1768, 9812}, {1788, 20060}, {2095, 6909}, {2320, 5425}, {2403, 4498}, {2975, 5221}, {3052, 3315}, {3060, 3937}, {3075, 18477}, {3337, 3616}, {3338, 3622}, {3522, 5709}, {3550, 17449}, {3579, 3889}, {3666, 10987}, {3832, 18540}, {3868, 4188}, {3916, 16865}, {3927, 17531}, {4000, 16568}, {4253, 21372}, {4359, 5361}, {4393, 20367}, {4641, 14997}, {4757, 21842}, {4850, 16668}, {4858, 14211}, {4973, 5902}, {5057, 17728}, {5068, 7330}, {5180, 10072}, {5211, 20064}, {5220, 9342}, {5265, 7098}, {5422, 22129}, {5439, 16859}, {5535, 5731}, {5536, 9778}, {5770, 6839}, {6510, 17092}, {6762, 20052}, {6763, 9780}, {7004, 9539}, {7085, 7496}, {7171, 15683}, {7191, 18193}, {7293, 7492}, {7419, 23085}, {9782, 19854}, {10056, 16763}, {11010, 20057}, {11246, 11680}, {12704, 20070}, {13243, 19541}, {14986, 17437}, {15650, 17535}, {15934, 17549}, {16704, 17490}, {16816, 18206}, {16948, 17054}, {17018, 17596}, {17577, 18541}, {18391, 20067}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2, 9965, 17484}, {57, 3218, 2}, {57, 3928, 3306}, {63, 3306, 7308}, {63, 7308, 3219}, {89, 17595, 17013}, {244, 4650, 17127}, {982, 17126, 17024}, {2094, 5435, 5905}, {3218, 3219, 3928}, {3219, 3306, 2}, {3306, 3928, 3219}, {3928, 7308, 63}, {5435, 5905, 2}
= (6-9-13)-search numbers [-1.26979273393357613, -0.230871406169741114, 4.38655671722508116]
*** 2.2
P5 = Perspector of (ABC, A*B*C*) = X(1)X(6246) ∩ X(8)X(10738)
= (2 a^5-2 (b-c)^3 (b+c)^2-a^4 (2 b+3 c)+a^3 (-4 b^2+6 b c+c^2)+a^2 (4 b^3+b^2 c-6 b c^2+c^3)+a (2 b^4-6 b^3 c+b^2 c^2+6 b c^3-3 c^4)) (2 a^5+2 (b-c)^3 (b+c)^2-a^4 (3 b+2 c)+a^3 (b^2+6 b c-4 c^2)+a^2 (b^3-6 b^2 c+b c^2+4 c^3)+a (-3 b^4+6 b^3 c+b^2 c^2-6 b c^3+2 c^4)) : : (barys)
= lies on these lines: {1,6246},{8,10738},{21,11231},{84,10728},{952,1392},{1000,10947},{2320,6980},{2800,5560},{7319,12247},{10090,15079},{10698,21398},{12119, 20107}
=(6-9-13)-search numbers [1.28038232156109107, 1.32577037412643563, 2.13187776679172779]
*** 2.3
P6 = Orthologic center (A*B*C*, A'B'C') = X(1)X(3) ∩ X(3036)X(3814)
=a (2 a^6+22 a^4 b c-6 a^5 (b+c)+4 (b-c)^4 (b+c)^2-a (b-c)^2 (6 b^3-13 b^2 c-13 b c^2+6 c^3)+a^3 (12 b^3-19 b^2 c-19 b c^2+12 c^3)-2 a^2 (3 b^4+7 b^3 c-21 b^2 c^2+7 b c^3+3 c^4)) : : (barys)
= lies on these lines: {1,3},{3036,3814},{5690,20107},{6681,10283}
= (6-9-13)-search numbers [1.27707896514572770, 1.34909407245309214, 2.11725521706497945]
*** 2.4
P7 = Parallelogic center (A*B*C*, A'B'C') = MIDPOINT OF X(3) AND X(36)
= a^2 (-2 a^5+2 a^4 (b+c)+a^3 (4 b^2-6 b c+4 c^2)+a^2 (-4 b^3+b^2 c+b c^2-4 c^3)+(b-c)^2 (2 b^3+b^2 c+b c^2+2 c^3)-2 a (b^4-3 b^3 c+3 b^2 c^2-3 b c^3+c^4)) : : (barys)
= X[5]-2*X[6681], X[104]+3*X[13587], 2*X[140]-X[3814], 5*X[631]-X[5080], 7*X[3523]+X[20067], 3*X[3582]-X[10738], 3*X[4881]-X[6265], X[6882]+X[15326], X[22793]-2*X[22835]
= lies on these lines: {1, 3}, {5, 6681}, {20, 10598}, {24, 1878}, {30, 6713}, {104, 13587}, {140, 3814}, {182, 9037}, {214, 14988}, {355, 4188}, {404, 9956},{515, 12619}, {535, 549}, {631, 5080}, {912, 22935}, {953, 8697}, {993, 5123}, {2771, 18861}, {3523, 20067}, {3582, 10738}, {3838, 6914}, {4299, 6958}, {4881, 6265}, {4996, 5440}, {5057, 6875}, {5146, 7501}, {5303, 6940}, {5450, 6924}, {5587, 18515}, {5840, 15325}, {5886, 6950}, {6882, 15326}, {6906, 9955}, {6942, 18481}, {6961, 10526}, {6970, 18516}, {6971, 10483}, {7288, 10525}, {7508, 10165}, {7743, 10058}, {11499, 19537}, {15446, 17606}, {16371, 22758}, {22793, 22835}
= midpoint of X(i) and X(j) for these {i,j}: {3,36},{1385, 10225},{2077,22765},{6882,15326}
= reflexion of X(i) in X(j) for these {i,j}: {5,6681},{1385,18857},{3814,140},{5048,15178},{22793,22835}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,10246,5010},{3,16203,5217},{3,22765,2077},{36,2077,22765},{36,5172,5126},{5172,5204,36},{5450,6924,18480},{14792,14800,2646},{15326,21154,6882}
= (6-9-13)-search numbers [2.10353805376833861,2.03152294646097417,1.26336180184908686]
Best regards
Ercole Suppa
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