For any point \(B = (x, y) \in S(r)\), \(\triangle OBC\) has at most one obtuse angle. The valid regions for these angles are strictly disjoint:
- Obtuse at \(O\): Occurs iff \(OB\cdot OC < 0\). This simplifies to the half-plane \(x+y<0\), cleanly bisecting \(S(r)\). Calculable in \(\mathcal{O}(1)\) time.
- Obtuse at \(C\): Occurs iff \(CB\cdot CO < 0\). This simplifies to \(x +y > r / 2\), defining a triangular region in the first quadrant. Calculable in \(\mathcal{O}(1)\) time.
- Obtuse at \(B:\) By Thales’s Theorem, \(B\) must lie strictly inside the circle with diameter \(OC\). With center \((r/8,r/8)\) and \(R^2=r^2/32\), the boundary is \((x-r/8)^2+(y-r/8)^2<r^2/32\). Computable in \(\mathcal{O}(r)\) time.
Lastly, points collinear with \(O\) and \(C\) need to be subtracted explicitly from the totals in all three regions.