This is not a self contained answer, and mostly has links (with a short description of each) to what I googled (which, in turn, does answer the question, I believe).
The following MathForum year 2000 page discusses the same problem and attributes the (geometric construction, as opposed to algebraic) solution to:
Simon Lhuilier (1750-1840) proved that:
The sections of an arbitrary triangular prism include
all possible forms of triangles.
It refers to Section 73 of the book by Heinrich Dorrie,
100 great problems of elementary mathematics
available online here, also here but not all sections.
This section discusses (with constructions) the Pohlke-Schwarz theorem (link to encyclopedia of math) which is a stronger result: for quadrilaterals instead of triangles. It states:
The oblique image of a given tetrahedron can always be determined in such
manner that it is similar to a given quadrilateral.
As a step in the proof, Lhuilier's theorem is proven on p.305, Fig.86.
The problem that you posted is equivalent to the following problem: Given three parallel lines in space, to construct an equilateral triangle with one vertex on each line. If the three lines are in the same plane the solution is easy (using $60^\circ$ rotation), and could be found here (MSE problem), or here (UGA), or here (BrownU), or here (1915 Monthly problem 454).
Variations, in three dimensions, and when the lines were not required to be parallel could also be found in this paper by Ochonski (some of the constructions there seem pretty crowded to me, they may perhaps necessarily be complicated). More on Pohlke-Schwarz theorem here (paper by Sklenarikova and Pemova) and here (1915 paper by Emch).
The problem with three parallel lines (in a plane, or not in a plane) is stated as exercises 1 and 2 on p.37 in the book by Z. A. Melzak, Invitation to Geometry google books link.