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How do we verify Snell's law using Fermat's principle in refraction if we don't know point B (Image in the post)? (zerobytes.monster)

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The original post: /r/askscience by /u/Western_Flight5276 on 2024-10-29 18:21:38.

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In this image isn't Fermat's principle of shortest time only defined if the point B is already given/mentioned or calculated using one of Snell's laws. If we don't know the point B it's impossible to verify Fermat's principle and if we use Snell's laws (which was verified using Fermat's principle) doesn't it become paradoxical. How should we find point B without using Snell's laws or justify using Snell's laws with Fermat's principle if we don't know point B?

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