The two smaller triangles are similar to each other, and to the big triangle. This is fairly easy to see, the angles match.
What this means informally is that if we put $\triangle ABC$ in a Xerox machine, then with the suitable Reduce setting we get one of the other triangles, and with another Reduce setting, you get the other one.
If $\triangle BDA$ has incircle of radius $p$, and $\triangle BDC$ has incircle of radius $q$, then because one is a scaled version of the other, the hypotenuses of the two triangles are also in the ratio $p:q$. Say the hypotenuses are respectively $kp$ and $kq$.
By the Pythagorean Theorem, the hypotenuse of $\triangle ABC$ is $\sqrt{(kp)^2+(kq)^2}=k\sqrt{p^2+q^2}$.
So the hypotenuses of the three triangles are in the ratio $p: q: \sqrt{p^2+q^2}.$
The incircle radii are in the same ratio. So if the two little ones are $p$ and $q$, the big one is $\sqrt{p^2+q^2}$.