Answer
$BA=2\sqrt 2,BC=3\sqrt 2,AC=5\sqrt 2$
$BA+BC=AC=5\sqrt 2$
Work Step by Step
In a given plane, three or more points that lie on the same straight line are called collinear points. Two points are always in a straight line. In geometry, the collinearity of a set of points is the property of the points lying on a single line. A set of points with this property is said to be collinear. In general, we can say that points are aligned in a line or a row.
$A(1,1+d)$, $B(3,3+d)$, $C(6,6+d)$
using the distance formula, $d=\sqrt {(x_{2}-x_{1})^2+{(y_{2}-y_{1})^2}}$
the distance from $A$ to $B$ is $BA=\sqrt {(3-1)^2+{((3+d)-(1+d))^2}}=\sqrt 8=2\sqrt 2$,
the distance from $B$ to $C$ is $CB=\sqrt {(6-3)^2+{((6+d)-(3+d))^2}}=\sqrt {18}=3\sqrt 2$,
the distance from $A$ to $C$ is $CA=\sqrt {(6-1)^2+{((6+d)-(1+d))^2}}=\sqrt {50}=5\sqrt 2$
$2\sqrt 2+3\sqrt 2=5\sqrt 2$
$5\sqrt 2=5\sqrt 2$
Therefore $BA+BC=AC$, so the three points are collinear.
