\(ABCD\) is a square. How many squares in the same plane have two or more vertices in the set \(\{A, B, C, D\}\)?
+6252 Choosing 2 adjacent vertices gets us 2 squares. One is the original square ABCD.
There are 4 ways to choose adjacent vertices and this gets us a total of 5 unique squares.
Choosing diagonal vertices gets 2 squares each. There are 2 pairs of diagonal vertices.
This gets us a total of 2x2=4 unique squares.
Thus there are a total of 5 + 4 = 9 unique squares, one of which is the original ABCD
