**Question: **Consider three circular parks of equal size with centers at A^{1}, A^{2}, and A^{3} respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles A^{1}A^{2}A^{3}, B^{1}B^{2}B^{3}, and C^{1}C^{2}C^{3}, , as shown. Three sprinters A, B, and C begin running from points A^{1}, B^{1} and C^{1} respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.

Let the radius of each circular park be r, and the distances to be traversed by the sprinters A, B and C be a, b and c respectively. Which of the following is true?

\(b - a = c - b = 3\sqrt 3 r\) | |

\(b - a = c - b = \sqrt 3 r\) | |

\(b = \frac{{a + c}}{2} = 2\left( {1 + \sqrt 3 } \right)r\) | |

\(c = 2b - a = \left( {2 + \sqrt 3 } \right)r\) |

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