How to find the fourth vertex of a square seen at an angle, given the first three and the side length?

Question

Algorithms

Conventionally, there is a square lying on a horizontal surface. The camera sees him from an angle. From it I get three vertex coordinates. How can I calculate the fourth?

Answer 1

As a mathematical model for the camera, the central projection is usually taken. I am almost certain that the central projection of some plane is an affine transformation. For the square ABCD, the vector(!) equality AC = AB + AD is true. Under the Affine transformation, this equality will be preserved.

Answer 2

An unambiguous solution of the problem without knowledge of the relative position of the camera and the plane is fundamentally impossible. I will illustrate.
Black marks are points known to us, red marks are valid options for the location of the fourth corner at different camera positions.