Question 68
Let $ABCD$ be a convex quadrilateral for which the line segments $AC$ and $BD$ intersect perpendicularly at a point $X$. Suppose $AX = 5$, $BX = 6$ and $CX = 20$ are satisfied. Let us denote by $O_1, O_2, O_3$ and $O_4$, respectively, the circle with center at $A$ and radius $AX$, the circle with center at $B$ and radius $BX$, the circle with center at $C$ and radius $CX$ and the circle with center at $D$ and radius $DX$. Determine the value of $DX$ if there exists a circle which is tangent to all of the circles $O_1, O_2, O_3$ and $O_4$. Here we denote by $YZ$ the length of the line segment $YZ$.