A point $ D$ is chosen on side $ BC$ of a triangle $ ABC$ such that the inradii of triangles $ ABD$ and $ ACD$ are equal. Consider in these triangles the excircles touching sides $ BD$ and $ CD$, respectively. Prove that their radii are also equal.
Source: Russian 2007
Tags: geometry, geometric transformation, homothety, geometry proposed
A point $ D$ is chosen on side $ BC$ of a triangle $ ABC$ such that the inradii of triangles $ ABD$ and $ ACD$ are equal. Consider in these triangles the excircles touching sides $ BD$ and $ CD$, respectively. Prove that their radii are also equal.