If these centres lie on a line, why is the triangle isosceles?

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In a triangle \(ABC\), \(I\) is the incentre, \(H\) the orthocentre, and \(O\) the circumcentre. Prove that \(AI\) bisects \(\angle OAH\). Hence, or otherwise, prove that if \(O\), \(H\) and \(I\) are collinear the triangle must be isosceles.

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