Prove that the points $A(-2,1,3)$, $B(1,2-4)$ and $C(-2,2,1)$ form a triangle.

Question

I have points $A(-2,1,3)$, $B(1,2-4)$ and $C(-2,2,1)$.

Do points $A,B,C$ form a triangle?

If it does how can I prove.

Answer 1

The equation through $A$ and $B$ is $$\frac{x+2}{3}=\frac{y-1}{1}=\frac{z-3}{-7}$$ and the point $C$ does not satisfy the above equation. Thus, the three points do not belong in a straight line.

Answer 2

Hint:

Let $A(x_1,y_1,z_1), B(x_2,y_2,z_2)$. Then $$AB=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$$ Prove that

$AB+BC>AC$ and $AC+BC>AB$ and $AB+AC>BC$