Question

How do you locate the center of a dilation?

Answer 1

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.

Explanation:

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'`.