HYACINTHOS 29181

[Angel Montesdeoca]:

Let ABC be a triangle, P point in the plane and  A'B'C'  an inscribed triangle in ABC with centroid P.

Let F be the finite fixed point of the affine transformation that sends A, B, C on A', B', C'  respectively.

When A'B'C' moves, the locus of point F is a conic c(P).

c(P) degenerates into two secant lines at Q if and only if P lies on K656.  Q lies on K219.

If P lies on K656, then P'=h(G,-1/3)[P] lies on K219, and c(P) is the polar conic of the point P' in the cubic K656.

Pairs {P = X(i), Q = X(j)} (P on K656 and Q on K219) for {i, j}:  {2, 2}, {3081, 14401}, {6545, 1647}, {8027, 1646}, {8028, 6544}, {8029, 1648}, {8030, 1649}, {8031, 14434}, {23610, 1645}, {23616, 1650}.

Other pairs  {P, Q} (P on K656 and Q on K219 is not in the current edition of ETC)

* If P = X(8023), Q8032 =

= X(2)X(824) ∩ X(4809)X(14402)  

= (b^3 - c^3) (-2 a^3 + b^3 + c^3)^2 : :

= lies on the cubic K219 and these lines: {2, 824}, {4809, 14402} 


* If P = X(23611),  Q23611 = 

= X(2)X(647) ∩ X(684)X(2491)    

= a^4(b^2-c^2)(b^4+c^4-a^2(b^2+c^2))^2(-a^4+2b^2c^2+a^2(b^2+c^2)) : : 

= lies on the cubic K219 and these lines: {2, 647}, {684, 2491} 

* If P = X(23612),  Q23612  = 

= X(2)X(650) ∩ X(647)X(1962)     

 = a^2 (b - c) (b^2 + c^2 - a (b + c))^2 (-a^2 + 2 b c + a (b + c)) : : 

= lies on the cubic K219 and these lines: {2, 650}, {647, 1962} 

* If P = X(23613),  Q23613  

= X(2)X(216) ∩ X(1636)X(2972)   

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

= lies on the cubic K219 and these lines: {2, 216}, {1636, 2972} 

* If  P = X(23614), Q23614 =  

= X(2)X(92) ∩ X(7004)X(7117)  

 = a^2 (b - c)^2 (a^3 + b^3 + b^2 c + b c^2 + c^3 - a^2 (b + c) - a (b^2 + c^2))^2 (a^4 (b + c) - a^2 (b - c)^2 (b + c) - 2 b (b - c)^2 c (b + c) + a (b^2 - c^2)^2 - a^3 (b^2 + c^2)) : : 

= lies on the cubic K219 and these lines: {2, 92}, {7004, 7117} 

* If P = X(23615),  Q23615 =   

= X(2)X(7) ∩ X(11)X(1146)   

= (b - c)^2 (-a + b + c)^2 (2 a^2 - (b - c)^2 - a (b + c)) : :

= lies on the cubic K219 and these lines: {2, 7}, {11, 1146} 

= reflection of X(14477) in X(2) 

Angel Montesdeoca

http://amontes.webs.ull.es/otrashtm/HGT2019.htm#HG100719