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Geometry Level 5

Let \(\theta _{1}\), \(\theta_{2}\) and \(\theta_{3}\) be three distinct solutions to the equation \(\tan{\theta} = \dfrac{17}{100}\) such that \(\theta _{i} \in \left( 0 , 3\pi \right) \), for all \(i={1,2,3}\). Find the value of

\[\tan{\dfrac{\theta_{1}}{3}} \times \tan{\dfrac{\theta_{2}}{3}} + \tan{\dfrac{\theta_{2}}{3}} \times \tan{\dfrac{\theta_{3}}{3}} + \tan{\dfrac{\theta_{3}}{3}} \times \tan{\dfrac{\theta_{1}}{3}}\]


This is not my original problem.
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