If I have a circumference and a rectangle inscribed into it, is the diagonal of the rectangle equal to the diameter of the circumference?

Question

I know that, indirectly, I have considered that (what I typed on the title) and, it might be true, but I want to prove it. I have done this so far: Let´s suppose I have a circumference and four points in it, $A,B,C,D$, and because it is rectangle, (I thought I could say that because if I calculate both slopes I get to $m_2m_1=-1$) I made this consideration , and its to give the cordinates to those points: $$A(x_1, y_1) , B(x_2, y_1) , C(x_1, y_2) , D(x_2, y_2)$$

And that all made this figure:

I hope you get what I mean, because probably I am being redudant, anyway, thanks in advance.

Answer 1

Yes. Euclid Book III Proposition 31 says that an angle inscribed in a semicircle is a right angle, and that if chords of a circle meet on the boundary at a right angle then that angle is inscribed in a semicircle.

http://aleph0.clarku.edu/~djoyce/elements/bookIII/propIII31.html

Answer 2

Well, you have the proof nearly written. You write that $P$ is at an equal distance from $A, B, C$, and even $D$ ? How many points sharing this property with $P$ exist...?