HYACINTHOS 27769

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

Na, Nb, Nc = the NPC centers of IBC, ICA, IAB, resp.

N1, N2, N3 = the isogonal conjugates of Na,Nb, Nc wrt triangles IBC, ICA, IAB, resp.

(ie N1,N2,N3 are the X(54)'s of IBC, ICA, IAB, resp)

ABC and the (degenerated) triangle N1N2N3 are circumcyclologic

ie the circumcircles of AN2N3, BN3N1, CN1N2, ABC are concurrent
the circumcircles of N1BC, N2CA, N3AB, N1N2N3 [= line OI] are concurrent.

 

Cyclologic centers?

Note: We can continue but the cyclologic centers probably are not interesting

Denote:
Oa, Ob, Oc = the circumcenters of AN2N3, BN3N1, CN1N2,resp
O1,O2,O3 = the circumcenters of N1BC, N2CA, N3AB, N1N2N3, resp.

The triangles OaObOc, O1O2O3 are circumcyclologic.

[Peter Moses]:

Hi Antreas,

>Cyclologic centers? 

X(6584) & X(484).

>The triangles OaObOc, O1O2O3 are circumcyclologic. 

At X(110) and the slightly unwieldly:

a (a^18+a^17 b-6 a^16 b^2-6 a^15 b^3+14 a^14 b^4+14 a^13 b^5-14 a^12 b^6-14 a^11 b^7+14 a^8 b^10+14 a^7 b^11-14 a^6 b^12-14 a^5 b^13+6a^4 b^14+6 a^3 b^15-a^2 b^16-a b^17+a^17 c+5 a^16 b c-2 a^15 b^2 c-18 a^14 b^3 c-4 a^13 b^4 c+22 a^12 b^5 c+14 a^11 b^6 c-14 a^10 b^7 c-10 a^9 b^8 c+20 a^8 b^9 c-6 a^7 b^10 c-30 a^6 b^11 c+12 a^5 b^12 c+18 a^4 b^13 c-6 a^3 b^14 c-2 a^2 b^15 c+a b^16 c-b^17 c-6 a^16 c^2-2 a^15 b c^2+26 a^14 b^2 c^2+12 a^13 b^3 c^2-44 a^12 b^4 c^2-30 a^11 b^5 c^2+38 a^10 b^6 c^2+42 a^9 b^7 c^2-22 a^8 b^8 c^2-38 a^7 b^9 c^2+14 a^6 b^10 c^2+24 a^5 b^11 c^2-8 a^4 b^12 c^2-10 a^3 b^13 c^2+2 a^2 b^14 c^2+2 a b^15 c^2-6 a^15 c^3-18 a^14 b c^3+12 a^13 b^2 c^3+50 a^12 b^3 c^3+2 a^11 b^4 c^3-44 a^10 b^5 c^3-24 a^9 b^6 c^3-8 a^8 b^7 c^3+32 a^7 b^8 c^3+50 a^6 b^9 c^3-26 a^5 b^10 c^3-34 a^4 b^11 c^3+12 a^3 b^12 c^3-4 a^2 b^13 c^3-2 a b^14 c^3+8 b^15 c^3+14 a^14 c^4-4 a^13 b c^4-44 a^12 b^2 c^4+2 a^11 b^3 c^4+53 a^10 b^4 c^4+9 a^9 b^5 c^4-28 a^8 b^6 c^4-8 a^7 b^7 c^4+3 a^6 b^8 c^4+3 a^5 b^9 c^4+2 a^4 b^10 c^4-4 a^3 b^11 c^4+2 a b^13 c^4+14 a^13 c^5+22 a^12 b c^5-30 a^11 b^2 c^5-44 a^10 b^3 c^5+9 a^9 b^4 c^5+45 a^8 b^5 c^5+2 a^7 b^6 c^5-28 a^6 b^7 c^5+9 a^5 b^8 c^5+9 a^4 b^9 c^5-2 a^3 b^10 c^5+24 a^2 b^11 c^5-2 a b^12 c^5-28 b^13 c^5-14 a^12 c^6+14 a^11 b c^6+38 a^10 b^2 c^6-24 a^9 b^3 c^6-28 a^8 b^4 c^6+2 a^7 b^5 c^6+10 a^6 b^6 c^6-8 a^5 b^7 c^6+18 a^3 b^9 c^6-2 a^2 b^10 c^6-6 a b^11 c^6-14 a^11 c^7-14 a^10 b c^7+42 a^9 b^2 c^7-8 a^8 b^3 c^7-8 a^7 b^4 c^7-28 a^6 b^5 c^7-8 a^5 b^6 c^7+14 a^4 b^7 c^7-14 a^3 b^8 c^7-18 a^2 b^9 c^7+6 a b^10 c^7+56 b^11 c^7-10 a^9 b c^8-22 a^8 b^2 c^8+32 a^7 b^3 c^8+3 a^6 b^4 c^8+9 a^5 b^5 c^8-14 a^3 b^7 c^8+2 a^2 b^8 c^8+20 a^8 b c^9-38 a^7 b^2 c^9+50 a^6 b^3 c^9+3 a^5 b^4 c^9+9 a^4 b^5 c^9+18 a^3 b^6 c^9-18 a^2 b^7 c^9-70 b^9 c^9+14 a^8 c^10-6 a^7 b c^10+14 a^6 b^2 c^10-26 a^5 b^3 c^10+2 a^4 b^4 c^10-2 a^3 b^5 c^10-2 a^2 b^6 c^10+6 a b^7 c^10+14 a^7 c^11-30 a^6 b c^11+24 a^5 b^2 c^11-34 a^4 b^3 c^11-4 a^3 b^4 c^11+24 a^2 b^5 c^11-6 a b^6 c^11+56 b^7 c^11-14 a^6 c^12+12 a^5 b c^12-8 a^4 b^2 c^12+12 a^3 b^3 c^12-2 a b^5 c^12-14 a^5 c^13+18 a^4 b c^13-10 a^3 b^2 c^13-4 a^2 b^3 c^13+2 a b^4 c^13-28 b^5 c^13+6 a^4 c^14-6 a^3 b c^14+2 a^2 b^2 c^14-2 a b^3 c^14+6 a^3 c^15-2 a^2 b c^15+2 a b^2 c^15+8 b^3 c^15-a^2 c^16+a b c^16-a c^17-b c^17):: 

on lines {{1,2687},{1325,2771}}.

Best regards,
Peter Moses.