Geometriculus?

Geometry Level 5

Getting bored in my history class I started drawing some amazing triangles .

Starting from \(A\) to \(B\) then to \(C\) and then again to \(A\) (loop). Now extending \(CA\) to \(D\) such that \(CA=AD\). Joining \(DB\) and extending \(BD\) to \(E\) such that \(BD=BE\). Again joining \(C\) and \(E\) and extending \(CE\) to \(F\) such that \(CE=CF\) and so on indefinitely without lifting the pen (as shown in figure).

Denote the following symbols:

\(\bullet \) \(\gamma \) be the set containing lines parallel to \(AB\).

\(\bullet \) \(\beta \) be the set containing lines parallel to \(BC\).

\(\bullet \) \( \Delta \) be the set containing lines parallel to \(CD\) or \(CA\).

\(\bullet \) \(\Theta \) be the set containing lines parallel to \(DB\) or \(BE\).

As i was getting extremely bored i drew 59 lines (assume)

We further denote the following:

\[a = n(\gamma), b = n(\beta), c = n(\Delta), d = n(\Theta) \]

Now the monic quartic equation whose roots are \(a,b,c,d\) is "double derivated" (double differentiated) and it can be represented as \(g{ x }^{ 2 }+ex+f=0\).

Find the value of \( |g| +|e|+|f| +24 \).

Details and Assumptions

  • \(AB, BC ,CD,DE\) are also included in the corresponding sets.

  • Consider \(CD,DE,EF,FG,GH\) and so on single line segments instead of two line segments.

This is original and the incident did happen.
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