HYACINTHOS 29421

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

 

Denote:

 

Na, NB, Nc = the NPC centers of OBC, OCA, OAB, resp.

 

A'B'C' = the medial triangle of NaNbNc

 

D = the Poncelet point of ABCO = X(125)

 

The reflections of DA', DB', DC' in BC, CA, AB, resp. are concurrent.

 

Point?

 

 

[Ercole Suppa] 

 

Hi Antreas, 

 

the reflections of DA', DB', DC' in BC, CA, AB, resp. concur at point:

 

Q = X(30)X(74) ∩ X(1994)X(14731) =

 

= 2 a^18-7 a^16 b^2+8 a^14 b^4-2 a^12 b^6-4 a^10 b^8+8 a^8 b^10-8 a^6 b^12+2 a^4 b^14+2 a^2 b^16-b^18-7 a^16 c^2+20 a^14 b^2 c^2-20 a^12 b^4 c^2+10 a^10 b^6 c^2-8 a^8 b^8 c^2+6 a^6 b^10 c^2+6 a^4 b^12 c^2-12 a^2 b^14 c^2+5 b^16 c^2+8 a^14 c^4-20 a^12 b^2 c^4+14 a^10 b^4 c^4-3 a^8 b^6 c^4+12 a^6 b^8 c^4-23 a^4 b^10 c^4+22 a^2 b^12 c^4-10 b^14 c^4-2 a^12 c^6+10 a^10 b^2 c^6-3 a^8 b^4 c^6-20 a^6 b^6 c^6+15 a^4 b^8 c^6-12 a^2 b^10 c^6+10 b^12 c^6-4 a^10 c^8-8 a^8 b^2 c^8+12 a^6 b^4 c^8+15 a^4 b^6 c^8-4 b^10 c^8+8 a^8 c^10+6 a^6 b^2 c^10-23 a^4 b^4 c^10-12 a^2 b^6 c^10-4 b^8 c^10-8 a^6 c^12+6 a^4 b^2 c^12+22 a^2 b^4 c^12+10 b^6 c^12+2 a^4 c^14-12 a^2 b^2 c^14-10 b^4 c^14+2 a^2 c^16+5 b^2 c^16-c^18 : : (barys)

 

= (7 R^2-2 SW)S^4+(108 R^6+18 R^4 SB+18 R^4 SC-21 R^2 SB SC-39 R^4 SW-4 R^2 SB SW-4 R^2 SC SW+6 SB SC SW-5 R^2 SW^2+2 SW^3)S^2-63 R^4 SB SC SW+39 R^2 SB SC SW^2-6 SB SC SW^3 : : (barys)

 

= lies on these lines: {30,74}, {1994,14731}, {3258,23292}, {3575,16221}, {7667,16188}, {12370,16168}

 

= (6-9-13)  search numbers:  [2.8923427107898714543, 2.9515807842947285259, 0.2623349954927032986]

 

Best regards

Ercole Suppa