Two circles of equal radius $$r$$ are inscribed in a semicircular sector of radius '1' such that they touch each other externally. Then $$r =$$ ___________

Place the semicircle with centre $$O$$ at the origin and its diameter along the $$x$$-axis. By symmetry the two circles touch each other on the $$y$$-axis and both touch the diameter, so their centres are $$(\pm r, r)$$. Each circle touches the arc internally, so the distance of its centre from $$O$$ is $$1 - r$$, which gives $$\sqrt{2}\,r = 1 - r$$. Hence $$r = \frac{1}{\sqrt{2} + 1} = \sqrt{2} - 1 \approx 0.414$$.