Suppose $z_1$, $z_2$ and $z_3$ are distinct points in $\hat{\mathbb {C}}$ . Show that $z$ is on the circle passing through $z_1$, $z_2$ and $z_3$ if and only if [$z$, $z_1$, $z_2$, $z_3$] is real or ∞.
124015
Let $[z_1,z_2,z_3,z_4]$ denote the  of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized circle if and only if $[z_1,z_2,z_3,z_4]\in\mathbb{R}$.    I saw this statement claimed in , and wondered how one would go about showing it. I know of some facts we could work with:   First, the definition: A  in $\mathbb{C}$ is either a circle or a line in $\mathbb{C}$. I know that a Mobius transformation maps generalized circles to generalized circles. I know that if $f$ is a Mobius transformation and $z_j'=f(z_j)$, $j=1,2,3,4$, then $[z_1',z_2',z_3',z_4']=[z_1,z_2,z_3,z_4]$.   I have posted a potential proof as an answer below. Please let me know if you think it's complete. Thank you!
53536
These usually aren't too bad but I had difficulties thinking of what the set  $$\{z\in\mathbb{C}:|z+i|=2|z|\}$$ looks like in the complex plane. I got as far as  $$|z+i|=2|z|\Rightarrow \sqrt{(z+i)(\overline{z+i})}=2z\overline{z}\Rightarrow  \sqrt{z\overline{z}+i\overline{z}-iz+1}=2z\overline{z}\\ \Rightarrow z\overline{z}-i(z+ z\overline{z})+1=4z\overline{z}\Rightarrow 3z\overline{z}=1-i2\Re(z)$$  And rearranging a bit I guess I can see that $z\overline{z}+\frac{2i\Re(z)}{3}=1/3$. If there weren't that second term on the left, this would be clear, but I am not sure how to think about the dependence on the real part of $z$ here.   Having graphed in wolfram, I know this is just a circle of radius 2/3 centered at $i/3$, but I would appreciate help with the thought process.
74091
Problem: Show that if $(b,c)=1$, then for any integer $a$ we have $(a,bc)=(a,b)(a,c)$. [Hint: Prove that each member of the alleged equation divides the other.]  This is the way I proved it (without the hint): $(a,b)(a,c)=(a(a,b),c(a,b))=(a^2,ab,ac,bc)=(a^2,bc,a(b,c))=(a^2,a,bc)=((a^2,a),bc)=(a,bc)$  But how should I prove it using that hint or at least a similar proof.
124150
Let be $$ a_1,\ldots,a_r \in \mathbb{N} $$  I want to show that  $$  a_1 \Bbb Z \cap \dotsb \cap a_r \mathbb{Z} = {\rm lcm}(a_1, \ldots, a_r) \Bbb Z$$  ok,I know that $ {2}\Bbb Z \cap 3\mathbb{Z} = 6\mathbb{Z}. $ I struggle getting an idea of how to prove it correctly, maybe you guys can help!
34406
$A=\{z\in\mathbb{C}: z^{p^n}=1\;\mathrm{for\;some\;integer}\;n\geq1\}$. I have to prove:  1) Every proper subgroup of $A$ is cyclic,  2) if $B,C$ are subgroups of $A$ then $B\subseteq C$ or $C\subseteq B$,  3) for every $n\geq0$ there exists an unique subgroup of $A$ with $p^n$ elements.  Could you help me please?
73433
Suppose $z_1, .....z_{m+1}$ are distinct elements of $F$ and $w_1,....,w_{m+1} \in F$. Prove that there exists a unique polynomial $p \in P_m(F)$ such that $p(z_j) = w_j$ for j=1,...m+1.  Any ideas on where to start/go with this one. I'm having quite some trouble with it.
88441
I would like to show that show $|t|\leq \pi$ implies $ |e^{it}-1|\geq 2|t|/ \pi$. I tried to proof it geometrically but without success.
123548
Show that if $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not   ring-isomorphic to $n\mathbb{Z}$.   Can I get some help to solve this problem
109527
Given two circle $\omega_1$ and $\omega_2$ which intersect at points $X$ and $Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX$ and $PY$ meet $\omega_2$ again at $A$ and $B$ respectively. Prove that the circumcircles of all triangles $PAB$ have equal radius.    What I tried is not worth adding here.
6135
Let $\mathbb{C}^{\times}$ be the group of nonzero complex numbers under multiplication. Then $\mathbb{C}^{\times}$ is the direct product of the circle group $T$ of unit complex numbers and the group $\mathbb{R}^{+}$ of positive real numbers under multiplication.
56694
we are given $z^n=(z+1)^n=1$, $z$ complex number. we want to prove that $n$ is divisible by $6$. I showed that $|z|=|z+1|=1$. Hence $z$ is on the intersection of two unit circles, one centered at $(0,0)$ and the other one centered at $(-1,0)$. Then $z$ can take two values $z=\operatorname{cis}(2 \pi /3)$ or $z=\operatorname{cis}(4 \pi/3)$. Then from $z^n=1=\operatorname{cis}(2 \pi k)$, $k$ integer I showed that $n$ should be divisible by $3$. I am left to show that $n$ should be divisible by $2$. I think if I figure out a way to show that $\operatorname{arg}(z+1)=\pi/3$ or $-\pi/3$ then I would be done, by using $(z+1)^n=\operatorname{cis}(2\pi k)$. but how can I do this? Any ideas are welcome! Thanks in advance!
116481
A regular polygon P is inscribed in a circle $\Gamma$. Let A, B, and C, be three consecutive vertices on the polygon P, and let M be a point on the arc AC of $\Gamma$ that does not contain B. Prove that  $MA\cdot MC=MB^2-AB^2$  I inscribed the polygon P in the unit circle and let B=1. Past this, I'm not really sure how to proceed. It might be helpful to have M at 1 as well but given that the polygon would have the n-th roots of unity in it B seemed like a good choice. What would be a good way to proceed from here?
110934
We have to write the equation of a family of circles passing through two given points $(x_1,y_1)$ &amp; $(x_2, y_2)$   From these points , I only know one circle that is  $$(x_1-x)(x_2-x)+(y-y_1)(y-y_2)=0$$  But now how Can I write other equations .
113480
Let $\gamma$ be the circle $\{z \in \mathbb{C}: \lvert z\rvert=1  \}$. Suppose $f$ is a function analytic on an open set containing $\gamma$ and its interior and that $\lvert\, f(z)\rvert&lt;1$ for each $z$ on $\gamma$. Show that $f$ has exactly one fixed point inside $\gamma$   That is, there is exactly one $z$ in the open unit disk with $f(z)=z$.   Is this a result of Louville's theorem?   I don't know how to approach it.
5211
Show that if $x$, $y$, $z$ are integers such that $x^2 + y^2 = z^2$, then at least one of them is divisible by $2$, at least one is divisible by $3$, and at least one is divisible by $5$.  I know that $x,y,z$ are sides of a right-angled triangles. If I start putting values, then I can show, but have no clue about mathematical approach.  How do I solve it mathematically, using modular arithmetic?
29374
Let $f\in H(B(0,1))$, $f(0)=0$, $f(B(0,1))\subset B(0,1)$. Suppose there are 2 different points $z_1$,$z_2\in B(0,1)$, s.t. $|z_1|=|z_2|$, $f(z_1)=f(z_2)$. Show that $|f(z_1)|=|f(z_2)|≤|z_1|^2=|z_2|^2$.    I have used the Schwarz-Pick Proposition to prove that $|f(z_1)|=|f(z_2)|≤|z_1|=|z_2|$ directly, but that is not enough. Thank you for help!
101688
I've got the following set: $\{|z-a|=k|z-b|\}$, where z is a complex number, a an b are fixed, and $k&gt;0$,$k \ne 1$. I need to prove that this is a circle (called Apollonius circle). I also have to prove that this circle's radius is equal to $k|a-b||1-k^2|^{-1}$ and it's centre is $(a-k^2b)(1-k^2)^{-1}$. I don't know what to do, I've tried to work with analytic circle equation ($|z-c|^2=r^2$), substituting given radius and center, but it didn't work. I also tried to square both sides of the first given equation ($|z-a|=k|z-b|$), which usually works with complex number, but also didn't get any result... Can somebody show m how to solve this problem? I will be very grateful.