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

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Prove that the internal bisector of an angle of a triangle divides the opposite side in the ratio of the lengths of the containing sides. 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.