Real elliptic curves and cevian geometry

We study the elliptic curve $E_a: (ax+1)y^2+(ax+1)(x-1)y+x^2-x=0$, which we call the geometric normal form of an elliptic curve. We show that any elliptic curve whose $j$-invariant is real is isomorphic to a curve $E_a$ in geometric normal form, and show that for $a \notin \{0, -1, -9\}$, the points on $E_a$, minus a set of $6$ points, can be characterized in terms of the cevian geometry of a triangle.