# Where Line Meets Regular Icosahedron

Geometry Level 4

A circle of radius 2 on the $$xy$$-plane centered at $$(0, 0)$$ has a regular pentagon inscribed inside it, not necessarily oriented as shown. This pentagon defines a regular icosahedron. From one of its vertices a line with a vector direction $$(3, -1, -2)$$ intersects the $$x$$-axis at $$\left(3+\sqrt{3}, 0, 0\right)$$. The coordinates of this vertex is $$(x, y, z)$$.

The sum $$x+y+z$$ can be expressed as

$\dfrac{a+\sqrt{b}}{c} ,$

where $$a, b, c$$ are integers and $$b$$ is square-free. Find $$a+b+c$$.

Note See also Regular Icosahedron for more details. Given a regular pentagon, it defines a regular icosahedron in that both the regular pentagon and the regular icosahedron share the same $$5$$ vertices. This would be possible with one unique regular icosahedron on either side of the plane of the regular pentagon.

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