HYACINTHOS 29226

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

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

D = the Feuerbach point X(11)

 

(Oa), (Ob), (Oc) = the circumcircles of ANbNc, BNcNa, CNaNb, resp.

[concurrent at the cyclologic center (ABC, NaNbNc)]

 

Ra = the radical axis of (Ob), (Oc)
Rb = the racical axis of (Oc), (Oa)
Rc = the radical axis of (Oa), (Ob)

R1, R2, R3  = the parallels to Ra, Rb, Rc through D, resp.

 

The reflections of R1, R2, R3 in BC, CA, AB, resp. are concurrent.

Point?


[César Lozada]:

 

Q = (name pending)

 

= a*( 2*a^14-3*(b+c)*a^13-(11*b^2-14*b*c+11*c^2)*a^12+6*(b+c)*(3*b^2-4*b*c+3*c^2)*a^11+12*(2*b^4+2*c^4-(4*b^2-5*b*c+4*c^2)*b*c)*a^10-(b+c)*(45*b^4+45*c^4-8*(11*b^2-13*b*c+11*c^2)*b*c)*a^9-(25*b^6+25*c^6-7*(10*b^4+10*c^4-(15*b^2-16*b*c+15*c^2)*b*c)*b*c)*a^8+2*(b+c)*(30*b^6+30*c^6-(64*b^4+64*c^4-(85*b^2-96*b*c+85*c^2)*b*c)*b*c)*a^7+2*(5*b^8+5*c^8-(32*b^6+32*c^6-(49*b^4+49*c^4-(36*b^2-29*b*c+36*c^2)*b*c)*b*c)*b*c)*a^6-(b+c)*(45*b^8+45*c^8-(96*b^6+96*c^6-(94*b^4+94*c^4-(88*b^2-93*b*c+88*c^2)*b*c)*b*c)*b*c)*a^5+(3*b^8+3*c^8+(48*b^6+48*c^6+(22*b^4+22*c^4-(8*b^2+21*b*c+8*c^2)*b*c)*b*c)*b*c)*(b-c)^2*a^4+2*(b^2-c^2)*(b-c)*(9*b^8+9*c^8-(2*b^4+2*c^4+(7*b^2-16*b*c+7*c^2)*b*c)*(b+c)^2*b*c)*a^3-2*(b^2-c^2)^2*(b-c)^2*(2*b^6+2*c^6+(12*b^4+12*c^4+(9*b^2+10*b*c+9*c^2)*b*c)*b*c)*a^2-(b^4-c^4)*(b^2-c^2)^3*(b+c)*(3*b^2-8*b*c+3*c^2)*a+(b^2-c^2)^6*(b+c)^2) : : (barys)

 

= lies on this line: {36, 3065}

 

= [ -1.7935230513492510, -0.9789626547619207, 5.1461877281345940 ]

 

 

César Lozada