Question

Show that the points $A(1,1,1),B(1,2,3)$ and $C(2,-1,1)$ are vertices of an isosceles triangle.

Answer 1

Let $A(1,1,1),B(1,2,3),andC(2,-1,1)$

The distance $AB,BC$ and $AC$ are calculated as follows,

$AB=\sqrt{(1-1{)}^2+(2-1{)}^2+(3-1{)}^2}=\sqrt{5}$,

$BC=\sqrt{(2-1{)}^2+(-1-2{)}^2+(1-3{)}^2}=\sqrt{14}$ and

$AC=\sqrt{(2-1{)}^2+(-1-1{)}^2+(1-1{)}^2}=\sqrt{5}$

Since $AB=AC$, the given points are the vertices of an isosceles triangle