Question 1210196: How many squares in the plane have at least two points in the lattice below as vertices?
https://www.svgrepo.com/show/446596/four-dots-square.svg
Found 3 solutions by CPhill, ikleyn, greenestamps:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! ```
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Let's break down this problem.
**1. Lattice Points**
The given lattice points are:
* (0, 0)
* (1, 0)
* (0, 1)
* (1, 1)
**2. Squares with at Least Two Points as Vertices**
We need to count the squares that can be formed using at least two of these points as vertices.
* **Squares with All Four Points as Vertices:**
* There is only 1 such square, the one formed by the four given points.
* **Squares with Two Points as Vertices:**
* **Squares with Sides Parallel to Axes:**
* The square with side length 1 is the only such square we have already counted.
* **Tilted Squares:**
* Consider the segment connecting (0, 0) and (1, 1). This is a diagonal of a square.
* Consider the segment connecting (1, 0) and (0, 1). This is also a diagonal of a square.
* These diagonals form a square with area 2.
* The vertices of this square are:
* (0, 0), (1, 1), (0, 2), (-1, 1)
* (0, 0), (1, 1), (2, 0), (1, -1)
* (0, 1), (1, 0), (2, 1), (1, 2)
* (0, 1), (1, 0), (-1, 0), (0, -1)
* We are only concerned with the squares that have at least 2 points from the lattice as vertices.
* The square with the diagonal from (0,0) to (1,1) has (0,0) and (1,1) as vertices.
* The square with the diagonal from (1,0) to (0,1) has (1,0) and (0,1) as vertices.
* The square with the diagonal from (1,1) to (0,0) has (1,1) and (0,0) as vertices.
* The square with the diagonal from (0,1) to (1,0) has (0,1) and (1,0) as vertices.
* Thus we have one square with at least 2 vertices.
**3. Counting All Squares**
* **1 Square** with all four points as vertices.
* **1 Square** with two points as vertices and is tilted.
**Total Squares**
* 1 + 1 = 2
**Therefore, there are 2 squares in the plane that have at least two points in the lattice as vertices.**
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
How many squares in the plane have at least two points in the lattice below as vertices?
https://www.svgrepo.com/show/446596/four-dots-square.svg
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Looking at the picture under the link, I do not understand,
how does it relate to the text in the post.
The referred picture is only good to scare people around.
Answer by greenestamps(13200)  (Show Source):
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