## X(3066) (2ND LEMOINE HOMOTHETIC CENTER)

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears            f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + 3b2 - c2)(a2 - b2 + 3c2)
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Suppose X is a point, with isogonal conjugate X- 1. It is well known that if the pedal triangle of X is homothetic to the antipedal triangle of X- 1. The Lemoine homothetic center, X(1286), is the homothetic center when X = X(6), and X(3065) is the homothetic center when X = X(2). See also X(1285). (Peter Moses, Dec. 7, 2005)

In general is X = x : y : z (trilinears), then the homothetic center is given by

a(x + y cos C)(x + z cos B) : b(y + z cos A)(y + x cos C): c(z + x cos B)(z + y cos A).
This is a correction for trilinears given in TCCT, p. 188.

X(3066) lies on these lines: 2,1350    3,373    6,110    25,182    51,394    107,458    125,381

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.