HYACINTHOS 29378

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of O (medial triangle).

 

Denote:

 

A", B", C" = the midpoints of AN, BN, CN, resp.

 

A'A" intersects the NPC (N) again at A*.

B'B" intersects the NPC (N) again at B*

C'C" intersects the NPC (N) again at C*

 

ABC, A*B*C* are perspective.

 

Perspector?

 

 

[Ercole Suppa] 

 

 

Hi Antreas, 

 

the perspector is:

 

Q = X(5)X(20189) ∩ X(1173)X(3628)

 

= (a^4-3 a^2 b^2+2 b^4-2 a^2 c^2-3 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^4-3 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-3 a^2 c^2-3 b^2 c^2+2 c^4) : : (barys)

 

= 15 S^4 +(-4 R^4+7 R^2 SB+7 R^2 SC+3 SB SC-3 R^2 SW+2 SB SW+2 SC SW+SW^2)S^2 +2 R^4 SB SC+3 R^2 SB SC SW+SB SC SW^2 : : (barys)

 

= lies on these lines: {5,20189}, {1173,3628}, {1656,26862}

 

= barycentric quotient X(115)/X(11792)

 

= trilinear quotient X(1109)/X(11792)

 

= (6-9-13)  search numbers:  [0.0421622367593084073, 0.0371066993821918168, 3.5955157346001039650]

 

 

Best regards

Ercole Suppa