Let any point $D$ be chosen on the side $BC$ of the triangle $ABC$. Let the radii of the incircles of the triangles $ABC, ABD$ and $ACD$ be $r_1, r_2$ and $r_3$. Prove that $r_1 <r_2 + r_3$.
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Tags: geometry, geometric inequality, inradius
Let any point $D$ be chosen on the side $BC$ of the triangle $ABC$. Let the radii of the incircles of the triangles $ABC, ABD$ and $ACD$ be $r_1, r_2$ and $r_3$. Prove that $r_1 <r_2 + r_3$.