My cursory analysis suggests that there are hidden constructions invisible in the game but used for the verification (you can see a base64-encoded task file flying through the network). The actual verification happens in an Emscripten-powered C++ engine and it is pretty opaque to me---the only thing I can identify is a parser for the task file.

My best guess without the actual source code is as follows: The perfect algorithm would involve the equality between two functions returning constructible numbers [1], where each function would map from unspecified points (that you can move around with the hand tool) to the result points. As far as I concerned comparing two constructible numbers is possible but inefficient [2], and comparing two functions is probably much harder. However for the purpose of game, we can approximate the equality by randomly evaluating a difference between two functions to the reasonable accuracy and checking if it is close to zero. Theoretically exact real arithmetic would be preferable, but adaptive fixed-precision real arithmetic may be also possible as long as rounding errors are tracked.

[1] https://en.wikipedia.org/wiki/Constructible_number

[2] https://mathoverflow.net/a/211556