# (Color+beauti)ful Hexagon!

Geometry Level 4

As shown in the above figure, in a regular hexagon $$ABCDEF$$, the segments $$BD, BE, CA$$ are drawn.

$$BD$$ and $$CA$$ intersect at point $$Q$$.

$$BE$$ and $$CA$$ intersect at point $$P$$.

$$A(\triangle BPQ)$$ is colored $$\color{Green}{\text{green}}$$

$$A(\triangle BCD)$$ is colored $$\color{Red}{\text{red}}$$

Rest of the area is colored $$\color{Purple}{\text{purple}}$$

If the ratio of areas of colors

$$\color{Purple}{\text{purple}}:\color{Green}{\text{green}}:\color{Red}{\text{red}}=\color{Purple}{a}:\color{Green}{b}:\color{Red}{c}$$

such that $$a,b,c \in \mathbb{N}$$ , $$\text{gcd}(a,b,c)=1$$.

Find $$a+b+c$$

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