In the adjoining figure, two Quadrants are touching at $$B$$. $$CE$$ is joined by a straight line, whose mid-point is $$F$$.

The measure of $$\angle CED$$ is ------.

With the quadrant centred at $$A$$ of radius $$r_1$$ and the quadrant centred at $$D$$ of radius $$r_2$$ touching at $$B$$, requiring the midpoint $$F$$ of $$CE$$ to lie exactly on the second arc forces $$r_1 = r_2(\sqrt{2}-1)$$. Placing coordinates and computing the angle between $$EC$$ and $$ED$$ with this ratio gives $$\angle CED = 67.5^\circ$$.