As others in the thread have pointed out, Euclidean geometry is decidable, but the proof of that fact corresponds to a highly inefficient algorithm for actually checking whether two points are equal.
An alternative approach would be to "cheat" and do a numeric approximation. Constructible numbers (the coordinates of any point the player can make) are always the solution to a quadratic equation whose coefficients have been previously constructed. So, especially for constructions that aren't very "deep" (as would be found in this game), we can easily find the decimal approximation to high precision very quickly. In no time at all, this approach can determine with high confidence that the player's solution and a prototype solution are probably the same.
I would speculate that constructing a near-miss that is falsely deemed correct by this method would require an inordinately complex construction.
An alternative approach would be to "cheat" and do a numeric approximation. Constructible numbers (the coordinates of any point the player can make) are always the solution to a quadratic equation whose coefficients have been previously constructed. So, especially for constructions that aren't very "deep" (as would be found in this game), we can easily find the decimal approximation to high precision very quickly. In no time at all, this approach can determine with high confidence that the player's solution and a prototype solution are probably the same.
I would speculate that constructing a near-miss that is falsely deemed correct by this method would require an inordinately complex construction.