How do you locate the center of a dilation?

This question assumes a task of finding a center of dilation by something that is given. See below the details about what should be given and how it can be used.

If a dilation (or scaling) is given, it is assumed that its center and a factor are given, so we can construct an image of any point.

If center of dilation is point ##O## and factor is ##f!=0##, any given point ##A## is transformed by a dilation into point ##A’## such that (a) points ##O##, ##A## and ##A’## are on the same line; (b) if ##f>0##, points ##A## and ##A’## are on the same side from center ##O##; if ##f<0##, point ##O## is in between ##A## and ##A’##; (c) Lengths of segments ##OA’## and ##OA## relate to each other at factor ##|f|##, that is ##|OA’|/|OA| = |f|##

If these two parameters, the center and the factor, are not known, something must be given to determine them. The minimum required to determine them is a source and an image of two different points.

Assume we have two points ##A## and ##B## and their images ##A’## and ##B’## as a result of dilation. Since center of dilation ##O## must lie on the same line as points ##A## and ##A’##, we can construct this line ##A A’## and state that center ##O## is located on it. Analogously, center ##O## must lie on line ##BB’##. Let’s construct it as well. The intersection of these two lines (and they must intersect since we know that center ##O## belongs to both) is our center of dilation.

Incidentally, we can find a factor of dilation since we know the relative position of points ##O##, ##A## and ##A’## and can measure the length of segments ##OA## and ##OA’##.