If a circle is inscribed in a square, then the diagonal of the square contains a diameter of the circle.

Understanding the Relationship Between the Circle and the Square

When a circle is inscribed in a square, it means that the circle touches all four sides of the square, fitting perfectly within it. The diameter of the circle will be equal to the side length of the square because the circle reaches from one side of the square to the opposite side at its widest point.

Break Down the Problem

1. Square Properties: The side length of the square is denoted as $s$.
2. Circle Properties: The diameter of the circle is denoted as $d$, where $d = s$.
3. Diagonal of the Square: The diagonal $D$ of the square can be calculated using the Pythagorean theorem, given by: $D = s\sqrt{2}$

Relevant Concepts

1. The inscribed circle's radius $r$ is half of the diameter: $r = \frac{d}{2} = \frac{s}{2}$
2. The diagonal of the square being $D$ also plays into how it relates to the inscribed circle.

Analysis and Detail

1. The circle fits perfectly inside the square, implying that its diameter ($d$) is equal to the side length of the square ($s$).
2. The diagonal calculated as $D = s\sqrt{2}$ will stretch across the square, thus will indeed contain the diameter of the inscribed circle ($d = s$).
3. The diagonal $D$ shows that the diameter of the circle is just a segment within it.

Verify and Summarize

• The diameter of the circle is equal to the side length of the square: $d = s$.
• The diagonal captures this dimension, proving that it contains the circle's diameter.

Final Answer

Yes, the diagonal of the square contains a diameter of the circle. Specifically, if the side length of the square is $s$, the diameter of the circle inscribed within it is equal to $s$, confirming the relationship.