$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that: \[\frac{AD}{DI_a}\leq\sqrt{2}-1.\]
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Tags: geometry, trigonometry, LaTeX, geometry proposed
$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that: \[\frac{AD}{DI_a}\leq\sqrt{2}-1.\]