HYACINTHOS 28368

[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* parallelogic

Parallelogic center (A'B'C', A*B*C*) = Feuerbach point X(11)

2.4. Parallelogic center (A*B*C*, A'B'C') ?

 

[Peter Moses]:

Hi Antreas,

--------------------------------------------------------- 

1.1)  

= X(4)X(94)∩X(93)X(22823)

Barycentrics (a^2+b^2-c^2) (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (a^2-b^2+c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) (2 a^4-4 a^2 b^2+2 b^4-4 a^2 c^2+3 b^2 c^2+2 c^4) ::
Barycentrics Csc[3 A] (8 Cos[A]-Sec[A]) Sin[A]^2 ::   [major center] 
= lies on these lines: {4,94},{93,22823},{275,1141},{317,328},{1989,3087},{3520,5961},{5627,10152},{10733,15469}.
= {X(4),X(265)}-harmonic conjugate of X(6344).
= X(6149)-isoconjugate of X(21400).
= barycentric product X(94)X(21844).
= barycentric quotient X(i)/X(j) for these {i,j}: {1989, 21400}, {21844, 323}

 

--------------------------------------------------------- 

1.2) 

(a^2-b^2-c^2) (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (2 a^16-6 a^14 b^2+4 a^12 b^4+2 a^10 b^6-2 a^6 b^10-4 a^4 b^12+6 a^2 b^14-2 b^16-6 a^14 c^2+14 a^12 b^2 c^2-9 a^10 b^4 c^2-3 a^8 b^6 c^2+9 a^6 b^8 c^2+9 a^4 b^10 c^2-30 a^2 b^12 c^2+16 b^14 c^2+4 a^12 c^4-9 a^10 b^2 c^4+8 a^8 b^4 c^4-7 a^6 b^6 c^4+54 a^2 b^10 c^4-56 b^12 c^4+2 a^10 c^6-3 a^8 b^2 c^6-7 a^6 b^4 c^6-10 a^4 b^6 c^6-30 a^2 b^8 c^6+112 b^10 c^6+9 a^6 b^2 c^8-30 a^2 b^6 c^8-140 b^8 c^8-2 a^6 c^10+9 a^4 b^2 c^10+54 a^2 b^4 c^10+112 b^6 c^10-4 a^4 c^12-30 a^2 b^2 c^12-56 b^4 c^12+6 a^2 c^14+16 b^2 c^14-2 c^16) :: 

= lies on these lines: {}.

--------------------------------------------------------- 

1.3) 

= X(4)X(94)∩X(1511)X(22823)

Barycentrics 2 a^16-6 a^14 b^2+4 a^12 b^4+2 a^10 b^6-2 a^6 b^10-4 a^4 b^12+6 a^2 b^14-2 b^16-6 a^14 c^2+16 a^12 b^2 c^2-11 a^10 b^4 c^2-9 a^8 b^6 c^2+13 a^6 b^8 c^2+11 a^4 b^10 c^2-24 a^2 b^12 c^2+10 b^14 c^2+4 a^12 c^4-11 a^10 b^2 c^4+20 a^8 b^4 c^4-10 a^6 b^6 c^4-19 a^4 b^8 c^4+36 a^2 b^10 c^4-20 b^12 c^4+2 a^10 c^6-9 a^8 b^2 c^6-10 a^6 b^4 c^6+24 a^4 b^6 c^6-18 a^2 b^8 c^6+22 b^10 c^6+13 a^6 b^2 c^8-19 a^4 b^4 c^8-18 a^2 b^6 c^8-20 b^8 c^8-2 a^6 c^10+11 a^4 b^2 c^10+36 a^2 b^4 c^10+22 b^6 c^10-4 a^4 c^12-24 a^2 b^2 c^12-20 b^4 c^12+6 a^2 c^14+10 b^2 c^14-2 c^16 :: 

= lies on these lines: {4,94},{1511,22823}.

--------------------------------------------------------- 

2.1) 

= X(1)X(4744)∩X(2)X(7)

Barycentrics a (2 a^2-2 b^2+3 b c-2 c^2) :: 

Barycentrics  3-4 Cos[A] :: (major center)

= 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},{323,23140},{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},{19245,23169}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9965, 17484), (57, 3928, 3306), (63, 3306, 7308), (63, 7308, 3219), (89, 17595, 17013), (244, 4650, 17127), (982, 17126, 17024), (2094, 5435, 5905), (3218, 3219, 3928), (3306, 3928, 3219), (3928, 7308, 63).
= X(3635)-zayin conjugate of X(9).
= crossdifference of every pair of points on line {663, 4770}.
= barycentric product X(86)X(4084).
= barycentric quotient X(4084)/X(10).

 

--------------------------------------------------------- 

2.2) 

= X(1)X(6246)∩X(8)X(10738)

Barycentrics (2 a^5-3 a^4 b+a^3 b^2+a^2 b^3-3 a b^4+2 b^5-2 a^4 c+6 a^3 b c-6 a^2 b^2 c+6 a b^3 c-2 b^4 c-4 a^3 c^2+a^2 b c^2+a b^2 c^2-4 b^3 c^2+4 a^2 c^3-6 a b c^3+4 b^2 c^3+2 a c^4+2 b c^4-2 c^5) (2 a^5-2 a^4 b-4 a^3 b^2+4 a^2 b^3+2 a b^4-2 b^5-3 a^4 c+6 a^3 b c+a^2 b^2 c-6 a b^3 c+2 b^4 c+a^3 c^2-6 a^2 b c^2+a b^2 c^2+4 b^3 c^2+a^2 c^3+6 a b c^3-4 b^2 c^3-3 a c^4-2 b c^4+2 c^5) :: 

= lies on Feuerbach hyperbola and 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}.

--------------------------------------------------------- 

2.3) 

= MIDPOINT OF X(2077) AND X(8148)

Barycentrics a (2 a^6-6 a^5 b+12 a^3 b^3-6 a^2 b^4-6 a b^5+4 b^6-6 a^5 c+22 a^4 b c-19 a^3 b^2 c-14 a^2 b^3 c+25 a b^4 c-8 b^5 c-19 a^3 b c^2+42 a^2 b^2 c^2-19 a b^3 c^2-4 b^4 c^2+12 a^3 c^3-14 a^2 b c^3-19 a b^2 c^3+16 b^3 c^3-6 a^2 c^4+25 a b c^4-4 b^2 c^4-6 a c^5-8 b c^5+4 c^6) :: 

= X[36] - 3 X[10247], 2 X[6681] - 3 X[10283], 5 X[5048] - X[13528], X[5535] - 9 X[16200].

= lies on these lines: {1,3},{3036,3814},{5690,20107},{6681,10283}.
= midpoint of X(2077) and X(8148).

--------------------------------------------------------- 

2.4) 

Barycentrics a^2 (2 a^5-2 a^4 b-4 a^3 b^2+4 a^2 b^3+2 a b^4-2 b^5-2 a^4 c+6 a^3 b c-a^2 b^2 c-6 a b^3 c+3 b^4 c-4 a^3 c^2-a^2 b c^2+6 a b^2 c^2-b^3 c^2+4 a^2 c^3-6 a b c^3-b^2 c^3+2 a c^4+3 b c^4-2 c^5):: 

= 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}.

 

---------------------------------------------------------

Best regards,
Peter Moses.