What Center Is Most Crucial?

Geometry Level 4

A triangle \(PQR\) was drawn on a Cartesian plane such that there exists a point \(O \) for which the distance \(\overline {PO } = \overline{QO} = \overline{RO} \) is equal to 4. And two of the vertices of this triangle have coordinates \((1,3) \) and \((5,6) \).

Given that the sine of one of the interior angles of this triangle must always be a constant. Find this constant.

Give your answer to 3 decimal places.

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