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)S_B + z(1 - z)S_C = x(1 + x)F
z(1 - z)S_C + x(1 - x)S_A = y(1 + y)F
x(1 - x)S_A + y(1 - y)S_B = z(1 + z)F
x + y + z = 1,

where S_A = (b² + c² - a²)/2; S_B, S_C 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].