HYACINTHOS 26240

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

Which is the locus of P such that the centroid of NaNbNc lies on the Euler line of ABC ?

G lies on the locus: The centroid of NaNbNc is the N of ABC.

 

[César Lozada]:

Locus = This circum-quartic:

 

q4:  CyclicSum[ (b^2-c^2)*y*z*(a^2*y*z+2*(-a^2 +b^2+c^2)*x^2) ] = 0

 

through ETC’s X(2), X(4), X(1113), X(1114), X(1383), X(3431)

 

ETC pairs (P,G(P)): (2,5), (4,5)

 

G(X(1113)) = midpoint of X(381)X(1113)

= 3*SB*SC-S^2+K*(2*S)/(3*R) : : (barycentrics),  where K = 2*S*OH

= (-3*R*S+2*K)*X(3)+(3*R*S+K)*X( 4)

= Shinagawa coefficients: (-3*R*S+2*K, 9*R*S)

= on line {2,3}

= midpoint of X(i) and X(j) for these {i,j}: {3,10720}, {381,1113}

= reflection of X(1313) in X(547)

= [ -0.566274993712567, -1.43430106887852, 4.894999834383003 ]

 

G(X(1114)) = midpoint of X(381)X(1114)

= 3*SB*SC-S^2-K*(2*S)/(3*R) : : (barycentrics),  where K = 2*S*OH

= (-3*R*S-2*K)*X(3)+(3*R*S-K)*X( 4)

= Shinagawa coefficients: (-3*R*S-2*K, 9*R*S)

= on line {2,3}

= midpoint of X(i) and X(j) for these {i,j}: {3,10719}, {381,1114}

= reflection of X(1312) in X(547)

= [ 5.825012578689002, 4.94012612553335, -2.467890179779789 ]

 

G(X(1383)) = 3*(27*R^2-8*SW)*S^2+81*R^2*SB* SC+8*SW^3 : :  (barycentrics)

= on line {2,3}

= [ 1.677019269395816, 0.80307532732134, 2.310680361733075 ]

 

G(X(3431)) = (117*R^2-32*SW)*S^2-27*R^2*SB* SC : : (barycentrics)

= (117*R^2-32*SW)*X(3)+(45*R^2-1 6*SW)*X(4)

= Shinagawa coefficients: (11*E+128*F, 27*E)

= on line {2,3}

= [ 5.643556626837062, 4.75914885905990, -2.258849325212654 ]

 

César Lozada