HYACINTHOS 25429

[Antreas P. Hatzipolakis]:

[From Hyacinthos 10583

Re: Some remarks on a NPC concurrence.    

Antreas P. Hatzipolakis, Oct 2, 2004]

 Let ABC be a triangle, and A'B'C', A"B"C" its orthic, medial triangles, resp. If h is the perpendicular to the Euler line of ABC from the NPC center N  of triangle ABC, then the triangle A1B1C1 is the reflection of the triangle A'B'C' in the line h and the triangle A2B2C2 is the reflection of the triangle A''B''C'' in the line h.
 
 (1)  The triangles ABC and A1B1C1 are perspective, and the perspector is  W =X(2)X(5627) /\ X(30)X(50)
 
  W  = (SA/(((b^2+c^2) a^4-2 (b^4-b^2 c^2+c^4)a^2+ b^6-b^4 c^2-b^2 c^4+c^6) (4 SA^2-b^2 c^2)) : ... : ...)
 
  with (6-9-13)-search numbers (0.489839199429536,  0.418452266923778,  3.12488712814659).
  W lies on lines: {2, 5627}, {30, 50}, {94, 2071}, {186, 476}, {265, 2072}, {1141, 3153}.
 
  (2)  The triangles ABC and A2B2C2 are perspective, and the perspector is  X(5627)  =  Yiu refletion point

Paul Yiu introduced this point on New Year's Day, January 1, 2014. He noted that X(74) is the unique point whose reflections in the sidelines of triangle ABC are collinear and perspective to ABC. The perspector is X(5627).

(3)  The triangles A'B'C' and A2B2C2 are perspective, and the perspector is X(403) = inverse-in-nine-point-circle of X(4).

(4)  The triangles A''B''C'' and A1B1C1 are perspective, and the perspector is X(2072) = inverse-in-nine-point-circle of X(3).

Angel Montesdeoca