What is a geometric construction corresponding to elliptic curve addition for Sharygin-isosceles triangles?

Consider the elliptic curve defined by the cubic: $$ a^3 + a^2b + ab^2 + b^3 + (a^2 + ab + b^2)c - (a+b)c^2 - c^3 = 0 $$ in $\mathbb{P^2}$ with distinguished point $[1, -1, 0]$ as identity.

Recently I learned that the real points of the identity component of this curve naturally parameterise Euclidean triangles with a neat property. Specifically they are scalene triangles such that the triangle formed by intersecting angle bisectors with opposite sides is isosceles. Here's an example:

(The triangle in red, constructed from one internal and two external angle bisectors, is isosceles.)

Let's call this property $P$. See here for a fuller discussion, with more pictures and a little history.

This means that given two triangles satisfying property $P$, there is a naturally associated third: their sum under elliptic curve addition. My questions are:

1. Can we find a geometric construction for the sum of two of these triangles ?

2. Is there is a natural family of geometric objects parameterised by the non-identity component of the elliptic curve ?

Regarding the second question, as discussed here it seems like triangles are out but it seems plausible we could find something. E.g., an idea with the sort of flavour I have in mind is as follows: a triangle satisfying property $P$, has a distinguished side. We can regard the other two sides as a singular quadric. Perhaps admitting non-singular quadrics gives us room to find an interpretation for points on this other component.

This is very nice! For now, I only have a little note to add that is too long for a comment. You can bring the curve $$C: a^3 + a^2b + ab^2 + b^3 + (a^2 + ab + b^2)c - (a+b)c^2 - c^3=0$$ to Weierstrass form $$E: y^2 + 1/3xy = x^3 + 7/9x^2 + 5/27x + 1/81$$ via $C\to E$ with coordinates $$(a/3 + b/3, -a/9,-2a - 2b - c).$$ And then you can easily bring $E$ to a minimal model $y^2 + xy = x^3 + x^2 - 2x$, and compute its Mordell-Weil group: it has rank $1$ and torsion $\mathbb{Z}/2\mathbb{Z}$. The group of rational points in the minimal model is generated by the point $(0,0)$ of order $2$ and the point $(2,2)$ of infinite order. (All this I have done with Magma.)