= Solution
The <reciprocal lattice> consists of all <wavevectors> $\mathbf G$ whose plane-wave phase is unchanged under every direct-lattice translation $\mathbf R$:
$$
e^{i\mathbf G\cdot\mathbf R}=1,\qquad
\mathbf G\cdot\mathbf R\in2\pi\mathbb Z.
$$
It expresses periodicity in <wavevector> space, with dimensions of inverse length.
Choose incident field convention $e^{i\mathbf k_i\cdot\mathbf r-i\omega t}$. The <wave> at the second atom acquires incident phase $\mathbf k_i\cdot\mathbf r_n$ relative to the atom at the origin. Its path to a distant detector is shorter by approximately $\hat{\mathbf k}_f\cdot\mathbf r_n$, contributing outgoing phase $-\mathbf k_f\cdot\mathbf r_n$. Therefore
$$
\Delta\phi=(\mathbf k_i-\mathbf k_f)\cdot\mathbf r_n
=\mathbf k_s\cdot\mathbf r_n.
$$
Neglecting variation of detector distance across the specimen, identical atoms have a common scattering amplitude factor and
$$
\boxed{S\propto\sum_ne^{i\mathbf k_s\cdot\mathbf r_n}.}
$$
The original geometry diagram shows the incident direction, outgoing direction and direct-space <displacement> separately; $\mathbf k_s$ is a <wavevector> difference, not an extra ray path.
The primitive reciprocal vectors obey
$$
\boxed{\mathbf A\cdot\mathbf a=2\pi,\qquad
\mathbf A\cdot\mathbf b=\mathbf A\cdot\mathbf c=0,}
$$
with cyclic versions for $\mathbf B,\mathbf C$. If $\mathbf k_s=h\mathbf A+k\mathbf B+l\mathbf C$, every <lattice> term has phase $2\pi$ times an integer, so all terms add constructively. Conversely, full translational reinforcement requires $\mathbf k_s\cdot\mathbf a$, $\mathbf k_s\cdot\mathbf b$ and $\mathbf k_s\cdot\mathbf c$ to be integer multiples of $2\pi$, hence $\mathbf k_s$ is reciprocal. These are the <Laue equations>. A finite <crystal> gives finite-width maxima rather than ideal reciprocal-space delta peaks; elastic scattering must also satisfy $|\mathbf k_i|=|\mathbf k_f|$.
For the two-dimensional <Bravais lattice> interpretation, with one atom per primitive cell, take
$$
\mathbf a=a(1,0),\qquad
\mathbf b=a(1/2,\sqrt3/2).
$$
Solving the duality conditions gives
$$
\mathbf A=\frac{2\pi}{a}(1,-1/\sqrt3),\qquad
\mathbf B=\frac{2\pi}{a}(0,2/\sqrt3),
$$
so
$$
\boxed{\mathbf G_{hk}=\frac{2\pi}{a}
\left(h,\frac{2k-h}{\sqrt3}\right),\qquad
|\mathbf G_{hk}|^2=\frac{16\pi^2}{3a^2}(h^2-hk+k^2).}
$$
The reciprocal primitive vectors have equal length $4\pi/(\sqrt3a)$ and angle $120^\circ$; the array itself is triangular. A different primitive basis can use a $60^\circ$ angle without changing the <lattice>. The direct primitive-cell area is $\sqrt3a^2/2$ and its reciprocal-cell area is $(2\pi)^2$ divided by that area.
For nonzero integer pairs, $h^2-hk+k^2$ has minimum one, attained for six shortest vectors. Therefore \b[the first nonforward maximum has $|\mathbf k_s|=4\pi/(\sqrt3a)$]. If the forward transmitted maximum is included literally, the smallest value is \b[$|\mathbf k_s|=0$]; the nonzero value is the first <diffraction> shell. Reaching that shell in elastic scattering additionally requires $4\pi/(\sqrt3a)\leq2|\mathbf k_i|$.
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