Uniquity of an Equilateral

Geometry Level 4

Two lines \(l_1\) and \(l_2\) intersect at an angle of \(\theta\). A point \(P\) is randomly chosen in the same plane as the two lines. If there is exactly one equilateral triangle with one vertex as \(P\), one vertex on \(l_1\), and one vertex on \(l_2\), then find the sum of all possible values of \(\theta\).

Details and Assumptions

\(0\le \theta \le 90^{\circ}\)

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