# 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.

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