# How can you intersect a cube at a single point?

Geometry Level 4

In the $$(x, y, z, w)$$ coordinate system, there is a line through the points $$(1, 3, -8, 4)$$ and $$(9, 8, 5, 6)$$. If the point of intersection of the line with the $$zwx$$ 3D space/cube/whatever you call it (point, line, plane, ??? ) can be represented as the point $$\left( -\dfrac{a}{b}, \dfrac{c}{d}, -\dfrac{e}{f}, \dfrac{g}{h} \right)$$, where each pair in parentheses $$(a, b), (e, f), (g, h)$$ contain only coprime positive integers, find the minimum sum of $$a+b+c+d+e+f+g+h$$.

Bonus: Make sense of the problem from a geometric point of view.

Notes and Assumptions:

$$e$$ here is a variable and isn't equal to $$\displaystyle \lim_{n \to \infty} \left( 1 + \dfrac{1}{n} \right)^n$$ or $$\displaystyle \sum_{k=0}^{\infty} \dfrac{1}{k!}$$

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