Finding all quadrilaterals in a set of intersections

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I want to take all the intersections of a set of lines and find all the convex quadrilaterals they create. I am not sure if there is an algorithm that works perfect for this, or if I need to loop through and create my own.

I have an array of lines, and all their intersections.

Lines and intersections:

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Example Quadrilaterals 1:

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Example Quadrilaterals 2 enter image description here

In this case, I would come out with 8 quadrilaterals.

How can I achieve this? If there isn't an algorithm I can implement for this, how can I check each intersection with other intersections to determine if they make a convex quadrilateral?

theory math language-agnostic geometry
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