What Is the Intersection of All Diameters in a Circle?

Thread starter Townsend
Start date Oct 24, 2004
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Elementary Set Set theory Theory

AI Thread Summary

The intersection of all diameters in a circle is the center of the circle, as it is the only point that all diameters share. To prove this, one could consider that each diameter spans from one point on the circle to another, bisecting the circle at the center. Since every diameter passes through the center, it confirms that the center is the common intersection point. Further exploration of geometric principles could strengthen this proof. The discussion emphasizes the importance of understanding the properties of circles and their diameters.

Oct 24, 2004

Townsend

Let C be a circle and let D be the set of all diameters of C. What is \capD?

I think it is the center of the circle since that would be the only point of intersection of all the diameters of the circle. Could someone let me know if I am correct?

Regards

Jeremy

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Oct 24, 2004

Hurkyl

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Do you have any thoughts as to how you could go about proving it?

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...

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