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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