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condition for angle AIO < 90

Source: France TST 2002; CHKMO 1999

June 30, 2007
geometryincentercircumcircletrigonometrygeometry proposed

Problem Statement

Let ABC ABC be a non-equilateral triangle. Denote by I I the incenter and by O O the circumcenter of the triangle ABC ABC. Prove that ∠AIO≤π2 \angle AIO\leq\frac{\pi}{2} holds if and only if 2\cdot BC\leq AB\plus{}AC.
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