The two circles in the diagram are incircles of $\triangle{ADB}$ and $\triangle{ADC}$. These incircles are tangent to $AD$ and each other at $G$.
i) If $AB=c$,$AC=b$ and $BC=a$, find the length of $BD$ in terms of $a,b$ and $c$.
ii) Let the radii of the two circles in the diagram be $r$ and $s$.Show that the length of $DE$ is $\sqrt{rs}$.
Efforts made: Working on the first part of the problem i tried to relate the tangents of the incircles to the legs of triangle $\triangle{ABC}$ in order to get an expression for $BD$ but as far as i try i always get something like $BD=2a + 2b -c +x$ where $x$ is some tangent to one of the two incircles.
Note: If you can give me only hints,it would be best.thanks in advance
Let $O_1,O_2$ be the incircle of $\triangle{ADB},\triangle{ADC}$ respectively. Also, let $H$ be the point on $O_1$ tangent to $AB$, and let $I$ be the point on $O_2$ tangent to $AC$.
For i) Let $AH=x,BH=y,CI=z,DE=w$. Then, since $AG=AI=x,BE=y,CF=z,DG=DF=w$, we have $$a=y+2w+z,\quad b=x+z,\quad c=x+y.$$
So, $BD=y+w=\cdots$
For ii) Each of $\angle{O_1GD},\angle{O_2GD},\angle{O_1ED},\angle{O_2FD}$ is equal to $90^\circ$. So, $O_1E$ is parallel to $O_2F$. And $O_1+O_2=r+s,EF=2w$. Consider the quadrilateral $O_1O_2FE$.