This is a follow up to this question: Constructing an equilateral triangle of a given side length inscribed in a given triangle
My attempts to solve the problem were initially focusing on algebraic/trigonometric. Two approaches led to two sets of equations:
Set 1 - unknowns $m, x_0, x_1$:
$(1+m^2)(x_0+x_1)^2=a^2$
$(x_0-a_0)^2(mx_0-b_0)^2=r_0^2$
$(x_1-a_1)^2(mx_1-b_1)^2=r_1^2$
Set 2 - unknown $x, y, z:
$y^2+(c-z)^2-2y(c-z)cos\alpha=s^2$
$x^2+(b-y)^2-2x(b-y)cos\gamma=s^2$
$z^2+(a-x)^2-2z(a-x)cos\beta=s^2$
I could not find a solution to these to sets of equation.
I successfully solved the original problem by mean of compass and a ruler.
As result two questions raised (at least for me):
Is there a NICE solution to any set of these equations?
Given the solution by construction, provides a solution to these two sets of equation. Is this a known practice to solve sets of equations? Any example in the literature?
Will be happy to clarify any issues...