HYACINTHOS 28392

 

[Antreas P. Hatzipolakis]:

 

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

Denote:

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

R1, R2, R3 =  the reflections of PA', PB', PC' in NbNc, NcNa, NaNb, resp. 

A*B*C* = the triangle bounded by R1, R2, R3

ABC, A*B*C* are parallelogic.

Parallelogic center (ABC, A*B*C*) = X(24144)  

Parallelogic center (A*B*C*, ABC) ?

[Peter Moses]:

Hi Antreas,

In gerneral, for P = {p,q,r}:

(A*B*C*, ABC):

(6 a^16-37 a^14 b^2+99 a^12 b^4-149 a^10 b^6+135 a^8 b^8-71 a^6 b^10+17 a^4 b^12+a^2 b^14-b^16-37 a^14 c^2+134 a^12 b^2 c^2-165 a^10 b^4 c^2+32 a^8 b^6 c^2+113 a^6 b^8 c^2-114 a^4 b^10 c^2+41 a^2 b^12 c^2-4 b^14 c^2+99 a^12 c^4-165 a^10 b^2 c^4+32 a^8 b^4 c^4+21 a^6 b^6 c^4+98 a^4 b^8 c^4-129 a^2 b^10 c^4+44 b^12 c^4-149 a^10 c^6+32 a^8 b^2 c^6+21 a^6 b^4 c^6-2 a^4 b^6 c^6+87 a^2 b^8 c^6-124 b^10 c^6+135 a^8 c^8+113 a^6 b^2 c^8+98 a^4 b^4 c^8+87 a^2 b^6 c^8+170 b^8 c^8-71 a^6 c^10-114 a^4 b^2 c^10-129 a^2 b^4 c^10-124 b^6 c^10+17 a^4 c^12+41 a^2 b^2 c^12+44 b^4 c^12+a^2 c^14-4 b^2 c^14-c^16) p+(a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-a^2 b^2 c^2-b^4 c^2-a^2 c^4-b^2 c^4+c^6) (2 a^10-7 a^8 b^2+10 a^6 b^4-8 a^4 b^6+4 a^2 b^8-b^10-7 a^8 c^2+6 a^6 b^2 c^2+7 a^4 b^4 c^2-13 a^2 b^6 c^2+7 b^8 c^2+6 a^6 c^4+3 a^4 b^2 c^4+6 a^2 b^4 c^4-18 b^6 c^4+4 a^4 c^6+11 a^2 b^2 c^6+22 b^4 c^6-8 a^2 c^8-13 b^2 c^8+3 c^10) q+(a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-a^2 b^2 c^2-b^4 c^2-a^2 c^4-b^2 c^4+c^6) (2 a^10-7 a^8 b^2+6 a^6 b^4+4 a^4 b^6-8 a^2 b^8+3 b^10-7 a^8 c^2+6 a^6 b^2 c^2+3 a^4 b^4 c^2+11 a^2 b^6 c^2-13 b^8 c^2+10 a^6 c^4+7 a^4 b^2 c^4+6 a^2 b^4 c^4+22 b^6 c^4-8 a^4 c^6-13 a^2 b^2 c^6-18 b^4 c^6+4 a^2 c^8+7 b^2 c^8-c^10) r : :

P = X(3), X(4): Hyacinthos 28391

P = N = X(5):

(A*B*C*, ABC): 

 
2 a^16-11 a^14 b^2+25 a^12 b^4-31 a^10 b^6+25 a^8 b^8-17 a^6 b^10+11 a^4 b^12-5 a^2 b^14+b^16-11 a^14 c^2+34 a^12 b^2 c^2-31 a^10 b^4 c^2-4 a^8 b^6 c^2+31 a^6 b^8 c^2-38 a^4 b^10 c^2+27 a^2 b^12 c^2-8 b^14 c^2+25 a^12 c^4-31 a^10 b^2 c^4-5 a^6 b^6 c^4+34 a^4 b^8 c^4-51 a^2 b^10 c^4+28 b^12 c^4-31 a^10 c^6-4 a^8 b^2 c^6-5 a^6 b^4 c^6-14 a^4 b^6 c^6+29 a^2 b^8 c^6-56 b^10 c^6+25 a^8 c^8+31 a^6 b^2 c^8+34 a^4 b^4 c^8+29 a^2 b^6 c^8+70 b^8 c^8-17 a^6 c^10-38 a^4 b^2 c^10-51 a^2 b^4 c^10-56 b^6 c^10+11 a^4 c^12+27 a^2 b^2 c^12+28 b^4 c^12-5 a^2 c^14-8 b^2 c^14+c^16 : : 

= lies on these lines: {2,16762},{3,16766},{5,195},{30,1141},{137,24147},{140,15345},{252,10126},{523,10096},{546,23337},{1154,12026},{3850,15307},{6592,10615},{7604,13469},{11016,16239},{13856,23280},{14051,20414}.
= midpoint of X(i) and X(j) for these {i,j}: {{5, 19553}, {137, 24147}, {1157, 1263}}.
= reflection of X(6592) in X(10614).
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195, 3459, 5), (16762, 16764, 2), (16766, 16768, 3).

Best regards,
Peter Moses.