# To circumscribe a square about a given circle

• astrololo
In summary, the conversation discusses the relationship between a line touching a circle at one point and forming a right angle with the diameter of the circle. The corollary of proposition 16 from book 3 is used to justify this relationship. However, there is some confusion about why Euclid goes on to prove this again in proposition 7 of book 4. It is clarified that the statement in proposition 7 uses the corollary, not proving it.
astrololo
http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV7.html

I was just wondering something . We know that if a line touches a circle at one point, then this means that this line is forming a right angle with the diameter of the circle. (“From this it is clear that the straight line drawn at right angles to the diameter of a circle from its end touches the circle.” According to corollary of proposition 16 from book 3)

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII16.html

In our construction, we would just draw the line at right angle to the diameter and with the corollary justify that is in fact touching the circle.

So, we already know that the angle made with the line touching the circle and the diameter is right, yet Euclid goes to prove it a second time by saying : “Then, since FG touches the circle ABCD, and EA has been joined from the center E to the point of contact at A, therefore the angles at A are right. For the same reason the angles at the points B, C, and D are also right.” There’s no problem with what he’s saying, but it seems a little repetitive to prove something which is already known. Maybe it’s me doing an error. Could someone to me what’s wrong here ? Thank you

But this statement:
astrololo said:
Then, since FG touches the circle ABCD, and EA has been joined from the center E to the point of contact at A, therefore the angles at A are right. For the same reason the angles at the points B, C, and D are also right.
uses the corollary, it does not prove it.

Not sure of understanding. I'm not saying that it proves it...

## 1. What does it mean to "circumscribe a square about a given circle"?

When we say to "circumscribe a square about a given circle," we are referring to the process of drawing a square that has its four corners touching the circumference of a given circle.

## 2. How can I circumscribe a square about a given circle?

To circumscribe a square about a given circle, you can use the Pythagorean theorem. The length of each side of the square will be equal to the diameter of the circle multiplied by the square root of 2.

## 3. What is the purpose of circumscribing a square about a given circle?

Circumscribing a square about a given circle can be useful in geometry and engineering, as it allows us to easily find the area and perimeter of the square and calculate the distance between the center of the circle and its corners.

## 4. Can any circle have a circumscribed square?

Yes, any circle can have a circumscribed square. However, the size of the square will vary depending on the diameter of the circle.

## 5. Are there any other methods for constructing a square around a circle?

Yes, there are other methods for constructing a square around a circle, such as using a compass and straightedge or using geometric constructions. However, the Pythagorean theorem is the most commonly used method for circumscribing a square about a given circle.

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