HYACINTHOS 26432

[César Lozada]:

Locus:

Let P be a point and A' the second intersection of AP and the circumcircle of PBC; similarly B',C'. 

 

The locus of P such that the Euler lines of A'BC, B'CA and C'AB concur at Q is:

 

Locus = {circumcircle} \/ { Gibert's Q003 through ETC's 1, 2, 4, 13, 14, 1113, 1114, 1156} \/ {a circum-sixtic through ETC's 13, 14}

 

 

 ETC pairs (P,Q) = (1, 100), (2,1296), (4, 110)

Notes: 

for P=X(13) or P=X(14), A'BC, B'AC and C'AB are equilateral

for P=X(1113) or P=X(1114), A'BC, B'AC and C'AB coincide with ABC

 

Q( X(1156) ) = a*(a-b)*(a-c)/(a^2-2*(b+c)* a+b^2+4*b*c+c^2) : :  (barycentrics)

= on the circumcircle and lines: {1,2291}, {56,3321}, {103,3576}, {104,1001}, {105,999}, {106,7290}, {692,2742}, {840,5126}, {972,3428}, {1308,1633}, {1477,13462}, {2371,5223}, {2717,5144}

= Trilinear pole of the line {6, 1155}

= [ 0.657437313474245, -0.71907206161246, 3.835050995266427 ]

 

César Lozada