find coordinates are of a parallelogram

Question

if the coordinates are (1, 2) (4, 3) (1, 0) (-2, -1), how can we can find whether it is of a parallelogram or not

Answer 1

Without drawing a graph, we know if it is a parallelogram using the following :

Four distinct points $(a,b),(c,d),(e,f),(g,h)$ are the vertices of a parallelogram if and only if either $$\frac{a+c}{2}=\frac{e+g}{2}\quad\text{and}\quad \frac{b+d}{2}=\frac{f+h}{2}$$ or$$\frac{a+e}{2}=\frac{c+g}{2}\quad\text{and}\quad \frac{b+f}{2}=\frac{d+h}{2}$$or$$\frac{a+g}{2}=\frac{e+c}{2}\quad\text{and}\quad \frac{b+h}{2}=\frac{f+d}{2}$$holds.