Randomly plotting the most likely angle

$ABCD$ is a square; and $P$ is a random point inside it. Let $\mathcal M$ denote the mode angle of $\theta$ .

If $\tan(\mathcal M) = \frac JK$ , where $J$ and $K$ are coprime positive integers, submit $J+K$ .

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Find the mean angle instead .

Find the median angle instead .