jmc

We may consider the diameter of circle $C$ as the base of the inscribed triangle; its length is $20\text{ cm}$. Then the corresponding height extends from some point on the diameter to some point on the circle $C$. The greatest possible height is a radius of $C$, achieved when the triangle is right isosceles: In this case, the height is $10\text{ cm}$, so the area of the triangle is $$\frac{1}{2}\cdot 20\cdot 10=\boxed{100}\text{ square centimeters}.$$