How to prove a circle consist of infinite points ?Proof using calculas or computational theory is appreciated?
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$\begingroup$Another proof using a parametrization:
The unit circle $S^1 = \{(x,y) \in \mathbb{R}^2 \mid x^2+y^2=1\}$ can be parametrized by $$\gamma: [0,2\pi) \to S^1,\ \gamma(t):=(\cos(t),\sin(t))$$ Note that $\gamma$ is a bijection (check!) and hence $S^1$ has the same cardinality as the interval $[0,2\pi)$ which again has the same cardinality as $\mathbb{R}$. Thus, the unit circle, as well as any other circle (you just need to modify the parametrization a bit), even contains uncountably many points.
There are more elementary proofs for this problem, but note that this proof via parametrizations can be used similarly for many geometric objects.
$\endgroup$ $\begingroup$For a function to be continuous btw a and b it has to cross all the point in between f(a) and f(b) where a and b are chosen such that f(a) and f(b) are distinct.
Hence from the property of real lines it has infinite points between f(a) and f(b) which is true for any continuous curve and hence is true for your case i.e., circle in general.
$\endgroup$ $\begingroup$Let $K$ be the center of a given circle.
Consider a line $l_1$ that passes through $K$ and intersects the circle at the points $A_1,A_1'$. Now consider the line $l_2\neq l_1$, perpendicular to $l_1$ at $K$ that intersects the circle at the points $A_2,A_2'$. It holds that $A_1,A_1'\neq A_2,A_2'$ because if they were identical, then we would have $l_1=l_2$.
Now notice that you can construct infinitely many different lines passing through $K$ and thus infinitely many points on the circle.
Indeed, for every $l_i,l_j$ we can construct another $l_k$, such that $l_k\neq l_i,l_j$.To do so let $\theta$ be the angle $A_iKA_j$ and let $Ka$ be its bisector. Then $Ka$ intersects the circle at a point $A_k\neq A_i,A_j$.
Similarly, consider as $\theta$ the angles $A_i'KA_j,A_i'KA_j',A_iKA_j'$ and cover the entirety of the circle with infinitely many such bisectors and thus infinitely different points.
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