HYACINTHOS 26287

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and ABCbCa, BCAcAb, CABaBc the three outer or inner Vecten squares.

Denote:

Oa, Ob, Oc = the circumcenters of ABaCa, BCbAb, CAcBc.

ABC, OaObOc are orthologic.

Orthologic centers?

[Peter Moses]:

Hi Antreas,

 

Inner: (Oa = {b^2+c^2+2 S,-b^2,-c^2})

(OaObOc, ABC) orthology: 

2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6+2 (a^2+b^2-c^2) (a^2-b^2+c^2) S:: 

on lines {{3,639},{4,6},{20,492},{30,59 1},{154,1585},{185,6291},{325, 489},{372,2794},{382,12601},{4 85,7694},{486,6399},{590,8414} ,{637,1350},{1132,3424},{1151, 6811},{1513,12963},{1586,1853} ,{3535,10192},{5085,7389},{ 5200,13567},{5921,12323},{6033 ,6230},{6459,7374},{6467,12298 },{6561,8721},{6813,9756},{ 7388,10516},{11381,12299}}.

midpoint of X(4) and X(5870).

3 X[4] - X[5871], 3 X[5870] + X[5871], 3 X[591] - 2 X[9733].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1588,5480),(4,6776,3070),(4 ,7582,6201),(4,10784,1587),(20 ,492,12305),(1587,10784,8550).

crosssum of X(3) and X(9733).

 

I assume the outer is the same with S->-S.

So 

Outer: (Oa = {b^2+c^2-2 S,-b^2,-c^2})

(OaObOc, ABC) orthology: 

2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6-2 (a^2+b^2-c^2) (a^2-b^2+c^2) S:: 

on lines {{3,640},{4,6},{20,491}, {30,1991},{154,1586},{185,6406 },{325,490},{371,2794},{382, 12602},{485,6222},{486,7694},{ 615,8406},{638,1350},{1131, 3424},{1152,6813},{1513,12968} ,{1585,1853},{3536,10192},{ 5085,7388},{5921,12322},{6033, 6231},{6460,7000},{6467,12299} ,{6560,8721},{6811,9756},{ 7374,13638},{7389,10516},{ 11381,12298}}.

midpoint of X(4) and X(5871).

3 X[4] - X[5870], 3 X[5871] + X[5870], 3 X[1991] - 2 X[9732].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1587,5480),(4,6776,3071),(4 ,7581,6202),(4,10783,1588),(20 ,491,12306),(1588,10783,8550), (6813,8982,1152).

crosssum of X(3) and X(9732).

 

(Note that the X(8982)'s "brother" with S->-S is not currently in ETC)

 

(ABC, OaObOc) orthology: X(4) for both cases.

 

 

Best regards,

Peter Moses.