= Solution
For an isotropic two-dimensional free-electron model, the occupied states form a disk centered at the reciprocal origin. Its <Fermi circle> first touches a zone side when its radius equals half the distance to a nearest reciprocal point:
$$
\boxed{k_F=\frac{|\mathbf b_1|}{2}
=\frac{\pi}{\sqrt2\,a}}.
$$
The contact points have coordinates $(\pm\pi/(2a),\pm\pi/(2a))$ with all four sign combinations. The distance to a zone vertex is $\pi/a$, which is larger; that distance would describe contact with a corner after the circle had already crossed a side.
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