|Information from Kimberling's Encyclopedia of Triangle Centers|
Trilinears (see below)
Barycentrics (see below)
A point P is an equilateral cevian triangle point if the cevian triangle of P is equilateral. Jiang Huanxin introduced this notion in 1997.
Jean-Pierre Ehrmann notes (11/6/02) that the normalized barycentric coordinates (x,y,z) of X(370) are the unique solution of this system:
y(1 - y)SB + z(1 - z)SC = x(1 + x)F
z(1 - z)SC + x(1 - x)SA = y(1 + y)F
x(1 - x)SA + y(1 - y)SB = z(1 + z)F
x + y + z = 1,
where SA = (b2 + c2 - a2)/2; SB, SC are defined cyclically, F = [2 area(ABC)]/sqrt(3), and x>0, y>0, z>0.
Jiang Huanxin and David Goering, Problem 10358* and Solution, "Equilateral cevian triangles," American Mathematical Monthly 104 (1997) 567-570 [proposed 1994].