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Length of the chord is given by  L =  2 R  Sin Ф,
           where R is the radius of the circle  and Ф is the angle subtended by the  chord at the center of the circle.

   If chord length is doubled then, Sin Ф' / Sin Ф = L' / L
         => Sin Ф' / Sin Ф = 2

       =>  Ф' = Sin⁻¹ [ 2 Sin Ф ]                          Ф' <= π/3  and Ф <= π/2

   for very small angles Ф,  the angle also nearly doubles.  Like for 15 degs or less.

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If a chord of a circle is doubled then the central angle is doubled.