60.1. Three distinct diameters are drawn on a unit circle such that chords are drawn as shown in Figure 2. If the length of one chord is $\sqrt{2}$ units and the other two chords are of equal lengths, what is the common length of these chords?
60.1. $\sqrt{2-\sqrt{2}}$ units
Refer to Figure 7. Let $\theta$ be the central angle subtended by the chord of length $\sqrt{2}$, and $\alpha$ the central angle subtended by each of the chords of equal lengths (and let $x$ be this common length). By the Law of Cosines, we have
$$
x^{2}=1^{2}+1^{2}-2 \cos \alpha=2-2 \cos \alpha .
$$

Since $\theta+2 \alpha=180^{\circ}$, we get
$$
\cos \alpha=\cos \left(90^{\circ}-\frac{\theta}{2}\right)=\sin \frac{\theta}{2}=\frac{\sqrt{2}}{2} .
$$

We can then solve for $x$.