Consider all quadrilaterals $ABCD$ such that $AB=10$, $BC=7$, $CD=6$, and $DA=9$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?

1. Since $AB+CD=10+6=16=BC+DA=7+9$, a quadrilateral with these sides can have an inscribed circle. As in the main problem, any circle inside has $r\le[ABCD]/s$, and the largest area comes from the cyclic quadrilateral, which is also tangential.
2. With $s=16$, Brahmagupta's area is\[\sqrt{(16-10)(16-7)(16-6)(16-9)}=\sqrt{6\cdot9\cdot10\cdot7}=\sqrt{3780}=6\sqrt{105}.\]
3. $r=\dfrac{6\sqrt{105}}{16}=\boxed{\dfrac{3\sqrt{105}}8}$.