Sadiku (2000), in developing the finite element method (using triangular elements as an example), defines a class of functions called element shape functions $\alpha_i(x,y)$, and claims that they are equal to $1$ when $i = j$, and $0$ otherwise:
I don't follow the reasoning. To me, the shape function doesn't depend on two parameters here to compare (e.g., $\alpha_{ij}$). In the first place, there's only $\{i, x, y\}$ to vary.
What does Sadiku mean by this?
With the reassurance that the properties claimed by Sadiku are indeed an intrinsic feature of barycentric coordinates as @fang explains, working out the algebra indeed leads to the right result. An abbreviated set of examples follows.
Off-point case:
$$ \alpha_1(x_2,y_2) = \frac{1}{2A} \left[ (x_2 y_3 - x_3 y_2) + (y_2 x_2 - y_3 x_2) + (x_3 y_2 - x_2 y_2) \right] \\ \implies \alpha_1(x_2, y_2) = \frac{1}{2A}(0) $$
On-point case:
$$ \alpha_1(x_1,y_1) = \frac{1}{2A} = \left[ (x_2 y_3 - x_3 y_2) + (y_2 x_1 - y_3 x_1) + (x_3 y_1 - x_2 y_1) \right] $$
Note that
$$ 2A = x_2 y_3 - x_2 y_1 - x_1 y_3 - x_3 y_2 + x_3 y_1 + x_1 y_2 $$
So
$$ \alpha_1(x_1,y_1) = \frac{x_2 y_3 - x_3 y_2 + y_2 x_1 - y_3 x_1 + x_3 y_1 - x_2 y_1}{x_2 y_3 - x_2 y_1 - x_1 y_3 - x_3 y_2 + x_3 y_1 + x_1 y_2} \\ = 1 $$
This is just the so-called barycentric coordinates of a point (x,y) with respect to the triangle defined by $V_{e1}$, $V_{e2}$ and $V_{e3}$. $\alpha_i=1$ when $i=j$ and 0 otherwise means that the shape function $\alpha_1$ has value 1 at $V_{e1}$ and value 0 at $V_{e2}$ and $V_{e3}$. Similarly, $\alpha_2$ has value 1 at $V_{e2}$ and value 0 at $V_{e3}$ and $V_{e1}$ and $\alpha_3$ has value 1 at $V_{e3}$ and value 0 at $V_{e1}$ and $V_{e2}$.