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Sine Rule and Incenter - RMO 2009 Geometry

The Problem

Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC.

Big Ideas

  • For any triangle ABC, \( \frac{\sin A}{a} = \frac{\sin B } {b} = \frac {\sin C }{c} \).
  • Addendo: If \( \frac{a}{b} = \frac{c}{d} \) then each of these ratios are equal to \( \frac {a+b}{c+d} \)
  • \( \sin (90 + \theta ) = \cos \theta \)
  • \( \cos ^2 2 \theta = 2 \cos ^2 \theta -1 = \cos ^2 \theta - \sin ^2 \theta \)

RMO 2009 problems, discussions and other resources. Read more

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