12. Which of all the isosceles triangles inscribed in a given semicircle has the greatest base, if one of the equal sides lies on the diameter, and the other is a chord?
12. Let the desired triangle be $ABC$ ($AC$ lies on the diameter) and in it $AB=AC=x, \quad BC=y$. By drawing $BD \perp AC$, we find that $y^{2}=2 x^{2}-2 x AD$, but $AD=\frac{x^{2}}{2 R}$, so $y^{2}=2 x^{2}-2 x \frac{x^{2}}{2 R}=\frac{1}{R} x^{2}(2 R-x)$.

The sum $x+(2 R-x)=2 R$ is a constant, therefore the product $x^{2}(2 R-x)$ has its maximum value when $\frac{x}{2}=\frac{2 R-x}{1}$. From this, $x=\frac{4}{3} R$.