# Why does the budget line cross the price offer curve instead of just touching it?

As far as I understand, the price offer curve shows all the utility-maximizing bundles generated by varying the price ratio but with the utility function remaining the same. However, in the image, the budget line is crossing the price offer curve, and the two black indifference curves seem like they don't belong to the green one and that they would also cross each other.

I'm probably getting some things wrong here, so I'd really appreciate it if someone could help :)

This is a little unclear to me, why would the utility function change? The price offer curve (of good $$i$$'s price) shows the utility-maximizing bundles generated by varying the price of good $$i$$.
There are three budget lines and three indifference curves in your graph, corresponding to different prices of good $$i$$. There is an optimal bundle for each; at that point, the indifference curve of the point is tangential to the budget line of the point. There is no guarantue that an indifference curve will be tangential to any of the other budget lines, it can freely intersect them.
Here is a Desmos where you can toggle the price of $$x$$ by sliding the red dot and observe movement along the price offer curve.