To circumscribe a square about a given circle

[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

 

[pbuk]

But this statement:

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.

 

[astrololo]

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