Question 1
Given a rectangle $ABCD$ with $AC$ length $e$ and four circles centers $A$, $B$, $C$, $D$ and radii $a$, $b$, $c$, $d$ respectively, satisfying $a + c = b + d \lt e$. Prove you can inscribe a circle inside the quadrilateral whose sides are the two outer common tangents to the circles center $A$ and $C$, and the two outer common tangents to the circles center $B$ and $D$.