Find sector centers in tic-tac-toe?

Question

Mathematics

There is a field, the width and height are constantly changing.
The field is divided into 9 squares or rectangles, like a keyboard on push-button phones or in a game of tic-tac-toe.
How to find the center of each sector?
More details can be found in the example from the forums about the anchor point, the center of an object or area, the rotation point.