# Zig-Zag Midpoint Wag

Geometry Level 5

From vertex $$C$$ of a unit equilateral triangle $$ABC,$$ we draw a line segment to the midpoint $$m_1$$ of the opposite side. From vertex $$m_1$$ of triangle $$BCm_1,$$ we similarly draw a line segment to the midpoint $$m_2$$ of the opposite side. Then we draw line segment $$m_2m_3$$ in triangle $$Cm_1m_2,$$ and so on. We travel indefinitely in this manner, getting closer and closer to a destination point we call $$P.$$

If $$\overline{AP} + \overline{BP} + \overline{CP}$$ can be expressed as $\dfrac{a}{b}\left(e\sqrt{c} + f\sqrt{d}\right)$ where $$a,b,c,d,e,f$$ are positive integers, $$\gcd(a,b) = \gcd(e,f) = 1,$$ and $$c$$ and $$d$$ are distinct and square-free, input $$a + b + c + d + e + f$$ as your answer.

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