how do you find the intersecting points of three $3d$ cones with origin and angle

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I'm working on a project and I'm a developer. the math is a bit, well, way beyond me. I can visualize things enough to see that they should work, but that's as far as my brain can take me on this one.

For this project, there are three points in a triangle. Each point has an origin for $xyz$, and an angle that'll make a cone. the direction will be towards one of the points in the triangle (based on some outside factors which one). Based on the project, these three cones should have a single intercept point between all three cones.

Here's a visualization of my goal

I've illustrated this by showing the red dots, which are the origins. The green dot would be the intercept point. My math skills aren't worldly enough to know where to look on how to solve something like this, so I'm hoping someone here and point me in the right direction!

For this, the size of the cone likely won't be more than $500$ ft, but for this application would stop at the intersection point and should be variable.

As for the intersection point itself, I'm trying to find a boundary intersection.