HYACINTHOS 28470

 

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of NBC, NCA, NAB, resp.

D = the Poncelet point of ABCN ( = X(137)

A'B'C' = the pedal triangle of D wrt triangle NaNbNc

Ma, Mb, Mc = the midpoints of A'Na, B'Nb, C'Nc, resp.

1. ABC, A'B'C' are homothetic. (Hyacinthos 28460)

2. ABC, MaMbMc are orthologic.

3. A'B'C', MaMbMc are orthologic

 

Orthologic centers?

 

[Peter Moses]:


Hi Antreas,

1).
X(25147).

2).
(ABC.  MaMbMc): X(25043).

(MaMbMc , ABC):

2 a^16-15 a^14 b^2+47 a^12 b^4-87 a^10 b^6+115 a^8 b^8-117 a^6 b^10+85 a^4 b^12-37 a^2 b^14+7 b^16-15 a^14 c^2+62 a^12 b^2 c^2-91 a^10 b^4 c^2+38 a^8 b^6 c^2+81 a^6 b^8 c^2-176 a^4 b^10 c^2+145 a^2 b^12 c^2-44 b^14 c^2+47 a^12 c^4-91 a^10 b^2 c^4+48 a^8 b^4 c^4-27 a^6 b^6 c^4+112 a^4 b^8 c^4-213 a^2 b^10 c^4+124 b^12 c^4-87 a^10 c^6+38 a^8 b^2 c^6-27 a^6 b^4 c^6-42 a^4 b^6 c^6+105 a^2 b^8 c^6-212 b^10 c^6+115 a^8 c^8+81 a^6 b^2 c^8+112 a^4 b^4 c^8+105 a^2 b^6 c^8+250 b^8 c^8-117 a^6 c^10-176 a^4 b^2 c^10-213 a^2 b^4 c^10-212 b^6 c^10+85 a^4 c^12+145 a^2 b^2 c^12+124 b^4 c^12-37 a^2 c^14-44 b^2 c^14+7 c^16 : : 

 

= X[137] + 3 X[547], 7 X[5] + X[1141], X[1263] + 7 X[3090], 5 X[1656] - X[6592], 15 X[1656] + X[11671], 3 X[6592] + X[11671], 3 X[1141] - 7 X[12026], 3 X[5] + X[12026], X[128] - 5 X[12812], 3 X[3628] - X[13372], 17 X[7486] - X[13512], 9 X[5055] - X[14072], X[930] - 9 X[15699], 11 X[5056] - 3 X[23237], X[140] + 3 X[23516], X[11671] - 9 X[25147], 5 X[1656] + 3 X[25147], X[6592] + 3 X[25147].

 

= lies on these lines: {5,49},{128,12812},{137,547},{140,23516},{930,15699},{1209,20413},{1263,3090},{1656,6592},{3628,13372},{5055,14072},{5056,23237},{7486,13512},{10096,15366},{14652,21308}.

 

{X(1656),X(25147)}-harmonic conjugate of X(6592).

3).
(A'B'C',  MaMbMc):

 

(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^22 b^2-8 a^20 b^4+31 a^18 b^6-80 a^16 b^8+154 a^14 b^10-224 a^12 b^12+238 a^10 b^14-176 a^8 b^16+85 a^6 b^18-24 a^4 b^20+3 a^2 b^22+a^22 c^2-10 a^20 b^2 c^2+34 a^18 b^4 c^2-51 a^16 b^6 c^2+4 a^14 b^8 c^2+156 a^12 b^10 c^2-390 a^10 b^12 c^2+526 a^8 b^14 c^2-429 a^6 b^16 c^2+206 a^4 b^18 c^2-52 a^2 b^20 c^2+5 b^22 c^2-8 a^20 c^4+34 a^18 b^2 c^4-36 a^16 b^4 c^4-35 a^14 b^6 c^4+96 a^12 b^8 c^4+24 a^10 b^10 c^4-370 a^8 b^12 c^4+665 a^6 b^14 c^4-576 a^4 b^16 c^4+248 a^2 b^18 c^4-42 b^20 c^4+31 a^18 c^6-51 a^16 b^2 c^6-35 a^14 b^4 c^6+106 a^12 b^6 c^6-61 a^10 b^8 c^6+93 a^8 b^10 c^6-406 a^6 b^12 c^6+727 a^4 b^14 c^6-565 a^2 b^16 c^6+161 b^18 c^6-80 a^16 c^8+4 a^14 b^2 c^8+96 a^12 b^4 c^8-61 a^10 b^6 c^8+16 a^8 b^8 c^8+85 a^6 b^10 c^8-420 a^4 b^12 c^8+682 a^2 b^14 c^8-376 b^16 c^8+154 a^14 c^10+156 a^12 b^2 c^10+24 a^10 b^4 c^10+93 a^8 b^6 c^10+85 a^6 b^8 c^10+174 a^4 b^10 c^10-316 a^2 b^12 c^10+602 b^14 c^10-224 a^12 c^12-390 a^10 b^2 c^12-370 a^8 b^4 c^12-406 a^6 b^6 c^12-420 a^4 b^8 c^12-316 a^2 b^10 c^12-700 b^12 c^12+238 a^10 c^14+526 a^8 b^2 c^14+665 a^6 b^4 c^14+727 a^4 b^6 c^14+682 a^2 b^8 c^14+602 b^10 c^14-176 a^8 c^16-429 a^6 b^2 c^16-576 a^4 b^4 c^16-565 a^2 b^6 c^16-376 b^8 c^16+85 a^6 c^18+206 a^4 b^2 c^18+248 a^2 b^4 c^18+161 b^6 c^18-24 a^4 c^20-52 a^2 b^2 c^20-42 b^4 c^20+3 a^2 c^22+5 b^2 c^22) : : 

 

= 3 X[547] - X[18807], X[25043] + 3 X[25147].

 

= lies on these lines: {137,1209},{547,18807},{3628,25150},{25043,25147}.

(MaMbMc, A'B'C') as (MaMbMc, ABC)

Best regards,
Peter Moses.