(Color+beauti)ful Hexagon!

Geometry Level 4

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

BDBD and CACA intersect at point QQ.

BEBE and CACA intersect at point PP.

A(BPQ)A(\triangle BPQ) is colored green\color{#20A900}{\text{green}}

A(BCD)A(\triangle BCD) is colored red\color{#D61F06}{\text{red}}

Rest of the area is colored purple\color{#69047E}{\text{purple}}


If the ratio of areas of colors

purple:green:red=a:b:c\color{#69047E}{\text{purple}}:\color{#20A900}{\text{green}}:\color{#D61F06}{\text{red}}=\color{#69047E}{a}:\color{#20A900}{b}:\color{#D61F06}{c}

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

Find a+b+ca+b+c

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