To my knowledge, the Delian problem does not exhibit a solution because the only instruments permitted to be used are a straight edge (a ruler with no measurements) and a compass. What is bothering me is, how is it proposed to mark root of 2 and other numbers on the triangles as suggested if root of 2 cannot be calculated with a straight edge and compass? And also, I must be missing something, but when I consider the dimensions of all the shapes involved, I do not land up with a cube... Have I missed something?

Regards,
Ian

https://www.cut-the-knot.org/Curriculum/Geometry/Delian.shtml

then I wonder whether you missed the title

What Is Wrong?

(Delian Problem Solved)

I believe that the title gives all the necessary indication that wht follows contains some kind of error. It does not say explicitly where.

sqrt(2) is constructible as is any other square root of an integer. This is just the diagonal of a unit'square. The cube root of 2, on the other hand, is not constructible and the Delian problem clearly contradicts this thesis.

Being able to construct the cube root of 2 would solve the Delian problem because the cube with that side would have the volume of 2 which is twice the volume of the unit cube.