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I have four points, one in each quadrant. How to prove whether it may be a parallelogram. What formula should I use?

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    If you have four points in each quadrant you have a total of sixteen points. Is this what you mean? – 2017-02-20

3 Answers 3

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Let's say you are given point $(x_1,y_1)$ in teh first quadrant,point $(x_2,y_2)$ int the second , point $(x_3,y_3)$ int the third and point $(x_4,y_4)$ in the fourth.Then it is enough to show that:

$\frac{y_1-y_2}{x_1-x_2}=\frac{y_4-y_3}{x_4-x_3}$

and,

$\frac{y_2-y_3}{x_2-x_3}=\frac{y_1-y_4}{x_1-x_4}$

NOTE: if $x_1=x_2$ then it must also be that $x_4=x_3$ and if $x_2=x_3$ the it must also be that $x_1=x_4$

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The points describe a parallelogram if the midpoints of the diagonal coincide. So for points $\newcommand{vP}[1]{\mathbf P_{#1}}\vP 1, \vP 2, \vP 3, \vP 4$ in their respective quadrants, we need to have $\dfrac{\vP 1 + \vP 3}{2}=\dfrac{\vP 2 + \vP 4}{2}$. We can simplify the test by dropping the division:

$$\text{Parallelogram} \iff \vP 1 + \vP 3= \vP 2 + \vP 4$$

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    it could be square. But need only a parallelogram. And here it is necessary to do something with the slope. – 2017-02-20
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    A square is a parallelogram. – 2017-02-20
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If centerpoints of diagonally opposite points coincide, then the figure formed joining them in the same order/ direction/sense is a parallelogram.