# Exploring the Intersection Area of Two Unit Squares

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## Chapter 1: Introduction to the Problem

Recently, I encountered a fascinating mathematical challenge that I felt compelled to share. The inspiration for this problem came from a Quora discussion highlighting some of the most captivating Math Olympiad problems (a link to this source is provided at the end of the article, though it includes spoilers!). While it may not strictly be an Olympiad question, it certainly possesses a charm of its own.

The problem can be succinctly stated. To clarify, when we mention the center of a square, we refer to the point where its diagonals intersect.

- Construct a square A with a side length of 1, centered at point c.
- Construct a second square B, also with a side length of 1, centered at the same point c.

**Goal:** Demonstrate that the area of the intersection between squares A and B is at least 3/4.

## Chapter 2: A Concise Solution Approach

A straightforward approach might involve calculating the exact area of the intersection by determining the area of the four identical triangles formed at the corners of square A where it intersects with square B. However, a more elegant solution can be found by reframing the problem as follows:

- Create square A with a side length of 1, centered at c.
- Make an identical copy of square A, designated as square B.
- Rotate square B around the center c to any desired angle.

**Goal:** Prove that the area of the intersection between squares A and B remains at least 3/4.

To enhance our analysis, we will draw the inscribed circle within square A.

The insight becomes clear once we draw the inscribed circle of square A; this circle is also the inscribed circle of square B due to its rotational symmetry about the center. Consequently, we can assert that the area of the intersection of squares A and B is at least equal to the area of the common inscribed circle. This circle has a radius of 1/2, leading to an area of at least π/4, which translates to a minimum area of 3/4.

This completes our proof!

## Chapter 3: Video Resources

To further understand this concept, here are two instructional videos:

In the first video, titled "Geometry Problem: Find the Area of the Intersection Between Two Squares," viewers will be guided through the process of determining the intersection area of two squares.

The second video, "Unit square, a circle and two quarter circles (A geometry puzzle)," presents an engaging geometry puzzle that complements the concepts discussed here.