ChinhLu

Assume that we have found B on the circle C1 (of center O and radius 2) and C on the circle C2 (of center O and radius 5) such that the area of the triangle 

ABC is maximum. Then OB must be perpendicular  to AC (otherwise we can move B to one of these two points on the circle C1). Arguing similarly, we see that OC is perpendicular to AB. Therefore, O must be the orthocenter of the triangle ABC. In particular, OA is perpendicular to BC. Suppose now that the coordinates of B and C are (-x,y_1) and (-x,y_2) respectively with $x\geq 0$. Then we need to maximise the following quantity

$$S(x):= (1+x) (\sqrt{2-x^2}+\sqrt{5-x^2}).$$

By standard analysis calculus, we can find $x=1$ and hence the position of B and C on the circles.