This question was raised when I was attempting to create a Python program inspired by Trilinear coordinates.

*Typo: I meant bijective.

Given:

The user is able to move any vertices of the triangle. Point P is fixed.

The returned coordinate is the distances from point P to the sides of the triangle made by the vertices.

User is allowed to make degenerate triangles. Examples: Points collinear, all points concurrent.

Question: Does this allow the user, in principle, to generate any coordinate in R^3?

Note:

If the vertices in a non-degenerate triangle are fixed and only P can move, then a bijection doesn't exist between distances and R^3. Consider P on a line defined by two vertices. There exists an upper and lower bound of distances possible between the third point and point P.

I don't think it matters if P can move or not. This in effect only shifts where the triangle should be to produce the desired coordinate.