Show love for triangles

Geometry Level pending

In an equilateral triangle with side length 1, consider dropping a perpendicular from a vertex onto the opposite side. Then, repeat this process, spiralling in clockwise, as in the picture. Where (in Cartesian coordinates, calling the bottom left vertex the origin) is the point to which this process converges?

suppose the coordinates comes as \((a,b)\), submit your answer as \(a^2+b^2\) to 2 decimal places.

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