This problem is fairly straightforward with a correct diagram. A hastily sketched diagram may not accentuate the relevant details. Segment ED is 25 units in length. The length of segment BC is 24 units. (how do we know?) To find the length of the altitude to A of triangle ABC, go 9/25 of the way from 9 to 16. Then you have the base and height of a triangle. The area comes to 138.24 square units.
The problem became more interesting when I tried to construct a diagram. A drawing will suffice if you know, for example, that the radii should be perpendicular to the tangents. The construction of a tangent to two circles (but not at their shared point) took me a few minutes to contrive. Also, triangle ABC is a right triangle for any two circles (if we do not keep the radii to 9 and 16) which are tangent in this way. (why?)
My students are always disgusted to learn that I do this stuff for fun.
This problem came from the Georgia Tech Math Olympiad Sample Test.