Question

The points $(X_1,Y_1)$, $(X_2,Y_2)$, $(X_1,Y_2)$ and $(X_2,Y_1)$ are always

• A. Collinear
• B. Concyclic
• C. Vertices of a square
• D. Vertices of rectangle

Answer 1

The correct option is A Collinear

As all the points lie on the same line this implies that area of the triangle formed by the points is zero.

So,using area formula in determinant form

$X_1\ast (Y_2-Y_3)-X_2\ast (Y_1-Y_3)+X_3\ast (Y_1-Y_2)=0$

$X_1\ast (Y_2-Y_3)+X_2\ast (Y_3-Y_1)+X_3\ast (Y_1-Y_2)=0$

Now,divide both sides by $X_1\ast X_2\ast X_3$

$(Y_2-Y_3)+X_2\ast (Y_3-Y_1)+X_3\ast (Y_1-Y_2)=0$

$\frac{Y_2-Y_3}{X_2-X_3}+\frac{Y_3-Y_1}{X_3-X_1}+\frac{Y_1-Y_2}{X_1-X_2}=0$

Its collinear.