If $AD=999$ and $PQ=200$, find the sum of the radii of those incircles.

Question

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle BDC=90° $. Let the incircles of $\Delta ABD $ and $ \Delta BCD $ touch $BD$ at $P$ and $Q$, respectively with $P$ between $Q$ and $B$. If $AD=999$ and $PQ=200$, find the sum of the radii of those incircles.

Answer 1

Let see the image. You have:

• the radii of the first incircle equals to $r_1 = EG = AG$.
• the radii of the second incicle equals to $r_2 = IP = DP$.

So, you have: $$AD = AG + GD = r_1 + DQ = r_1 + r_2 + PQ$$

Then, $$r_1 + r_2 = AD - PQ.$$