Tangling Circles

Geometry Level 5

Three circles of equal radius all intersect one another at the center of the bigger circle with twice the radius, and then the lines are drawn from the intersecting points on the big circle to form a triangle as shown above.

The triangle is then divided into 3 quadrilaterals by drawing the lines from the center of the big circle to the intersecting points of the small circles: red, blue, green areas (shown on the left). Alternatively, the triangle can be divided into 3 smaller triangles by simply drawing the lines from the center to the vertices: yellow, purple, cyan areas (shown on the right).

Given that the ratio of the area of the red region, the area of the dark blue region and the area of the green region be \(11:12:13\).

If the ratio of area of the yellow region, the area of the purple region and the area of the light blue region be \(a:b:c\), where \(\gcd(a,b,c) =1\), compute \(\dfrac{a+b+c}{3}\).

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