USAJMO 2010 Question 6

In triangle $ABC$, $\angle A=90^\circ$. Points $D\in AC$ and $E\in AB$ satisfy $\angle ABD=\angle DBC$ and $\angle ACE=\angle ECB$. Let $I=BD\cap CE$. Can all six lengths $AB,AC,BI,ID,CI,IE$ be integers? Prove your answer.

Proposed by Zuming Feng (credit in the supplied notes).