HYACINTHOS 28775

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of N..

 

Denote:

 

(Na), (Nb), (Nc) = the NPCs of NBC, NCA, NAB, resp.

 

(N1), (N2), (N3) = the reflections of (Na), (Nb), (Nc) in AN, BN, CN, resp.

 

R1, R2, R3 = the redical axes of ((N2), (N3)), ((N3), (N1)), ((N1), (N2)), resp.

 

A*B*C* = the triangle bounded by the reflections of R1, R2, R3 in BC, CA, AB, resp.

 

A'B'C', A*B*C* are parallelogic.

 

APH 

 

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

 

 

[Ercole Suppa]

 

Dear Antreas

 

 

*** 1. The parallelogic center (A'B'C', A*B*C*) is the point W1=X(10096)

 

 

*** 2. The parallelogic center (A*B*C*, A'B'C') is the point 

 

W2 = X(54)X(21394) ∩ X(186)X(323)

 

= a^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (a^16-4 a^14 b^2+5 a^12 b^4+a^10 b^6-10 a^8 b^8+14 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16-4 a^14 c^2+8 a^12 b^2 c^2-a^10 b^4 c^2-3 a^8 b^6 c^2-12 a^6 b^8 c^2+28 a^4 b^10 c^2-23 a^2 b^12 c^2+7 b^14 c^2+5 a^12 c^4-a^10 b^2 c^4-a^8 b^4 c^4-2 a^6 b^6 c^4-18 a^4 b^8 c^4+39 a^2 b^10 c^4-22 b^12 c^4+a^10 c^6-3 a^8 b^2 c^6-2 a^6 b^4 c^6+2 a^4 b^6 c^6-21 a^2 b^8 c^6+41 b^10 c^6-10 a^8 c^8-12 a^6 b^2 c^8-18 a^4 b^4 c^8-21 a^2 b^6 c^8-50 b^8 c^8+14 a^6 c^10+28 a^4 b^2 c^10+39 a^2 b^4 c^10+41 b^6 c^10-11 a^4 c^12-23 a^2 b^2 c^12-22 b^4 c^12+5 a^2 c^14+7 b^2 c^14-c^16) :: (barys)

 

= (17 R^2+SB+SC-5 SW)S^4 + (-64 R^6+8 R^4 SB+8 R^4 SC+68 R^4 SW-6 R^2 SB SW-6 R^2 SC SW-25 R^2 SW^2+SB SW^2+SC SW^2+3 SW^3+SB SC (-15 R^2+5 SW))S^2 + SB SC (30 R^6-40 R^4 SW+19 R^2 SW^2-3 SW^3) : : (barys)

 

= lies on these lines: {54,21394}, {186,323}, {6150,12006}

 

= (6-8-13) search numbers [3.21956868693878105, 2.73141771932135791, 0.263728205328616994]

 

 

Ercole Suppa