query: Is it possible to prove that in a circle $\frac{circumference}{diameter}$ is constant?
positive_passages.docid: 148233
positive_passages.text: Proof that every circle has the ratio of $\pi$
negative_passages.docid: 96539
negative_passages.text: is $f$ constant or dilation
negative_passages.docid: 17751
negative_passages.text: Find the greatest value of $\arg z$ achieved on a circle in $\mathbb{C}$
negative_passages.docid: 73867
negative_passages.text: Radius of circumscribed circle of triangle as function of the sides
negative_passages.docid: 104548
negative_passages.text: Geometric proof for irrationality of $\pi$
negative_passages.docid: 40596
negative_passages.text: Proving a relation between inradius ,circumradius and exradii in a triangle
negative_passages.docid: 120953
negative_passages.text: Show that, $2\arctan\left(\frac{1}{3}\right)+2\arcsin\left(\frac{1}{5\sqrt2}\right)-\arctan\left(\frac{1}{7}\right)=\frac{\pi}{4}$
negative_passages.docid: 53307
negative_passages.text: how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$
negative_passages.docid: 96388
negative_passages.text: Prove $\sin(x)\tan(x) > x^2$ for $x \in ( \,0, \frac{\pi}{2}) \,$
negative_passages.docid: 111094
negative_passages.text: Help with proving that $\pi$ is irrational
negative_passages.docid: 155241
negative_passages.text: How do we know the ratio between circumference and diameter is the same for all circles?
negative_passages.docid: 75679
negative_passages.text: How to prove that $2 \arctan\sqrt{x} = \arcsin \frac{x-1}{x+1} + \frac{\pi}{2}$
negative_passages.docid: 137711
negative_passages.text: Alternative proof that $[\Bbb{Q}(\zeta_n) : \Bbb{Q} ]= \varphi(n)$ uses circular reasoning?
negative_passages.docid: 61262
negative_passages.text: Is $\cot(z)-\frac1z$ bounded on a given circle?
negative_passages.docid: 14447
negative_passages.text: Direct proof that $\pi$ is not constructible
negative_passages.docid: 28399
negative_passages.text: Showing $f$ is constant
negative_passages.docid: 12158
negative_passages.text: Show that $\arctan\frac{1}{2}+\arctan\frac{1}{3}=\frac{\pi}{4}$