a) The equation of the circle centered at $1+2i$ with radius $1$ can be written as $[ z\overline{z} + pz + q \overline{z} + r = 0.]$What is the ordered triple $(p, q, r)$?  b) Let $A_1 A_2 \dotsb A_{11}$ be a regular 11-gon inscribed in a circle of radius 2. Let $P$ be a point, such that the distance from $P$ to the center of the circle is 3. Solve $PA_1^2 + PA_2^2 + \dots + PA_{11}^2.$  I have no clue how to get started on these problems.
37955
$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find $LK_1^2 + LK_2^2 + \dots + LK_{11}^2$.  Any suggestions as to how to solve this problem? I'm unsure what method to use.
3557
Let $\Gamma$ be a circle that passes through the origin. Show that we can find real numbers $s$ and $t$ such that $\Gamma$ is the graph of $r = 2s \cos (\theta + t).$  I know this has to be converted to a cartesian equation, but how do I do this, and what do I do after?  Thanks
57830
Given three vectors $u,v,w \in S^2$ and the triangle $[u,v,w]$ I want to find its circumscribed circle. However, I don't know how to approach this problem. Would some one please explain?  In my understanding, one needs to somehow find the altitudes of the spherical triangle and then the point of its intersection (circumcentre)? Then what about the radius?
116138
So I just learned that complex numbers cannot be ordered from .  Now, does this mean that 4 - 3i &lt; 4 + 4i isn't true? I haven't really seen any concrete examples and I don't even get what the proof means, but I know it makes sense - I just can't relate to it.  In short, what are some concrete examples of this theorem? What is also a walkthrough of the theorem?
25924
There is three square in a circle like the picture attached.   How can find radius of circle  ? I know that's not a hard problem,but I was away from geometry for years.  If possible give me hint or idea to start to solving.   Thanks in advance.   the smallest side is $6$ and medium square side is $6+6$ and big side is $6+6+6$ But I got stuck here and not to go over there ...
115277
Question: Tangents drawn from the point $P(1, 8)$ to the circle $x^2 + y^2 -6x -4y -11=0$ touch the circle at the points $A$ and $B$. What is the equation of the circumcircle of the triangle $PAB$?  I tried drawing a diagram, and found the length of the tangent. However, I lost as to where to proceed from there. Please help me!
119992
If $z$ lies on the circle $|z-1|=1, $ then the what is the value of $\frac{z-2}{z}$ ($z\ne 0 \ \text{and} \ 2$ ) is?  What I am aware of : the equation $|z-1|=1$ represent a circle with  centre $(1,0)$ and radius $1$. Also $$|z-1|=1 \implies (z-1)^2=1 \\ \implies  z^2-2z+1=1 \implies z(z-2) =0$$ $\implies z = 0; z =2 $ I think this is wrong.. please suggest thanks..
99702
Let $\Gamma$ denote the positively oriented circle of radius $2$ with center at the origin. Let $f$ be an analytic function on $\{z \in\mathbb{C} : |z|&gt;1 \}$, and let $$\lim_{z\to \infty}f(z)=0.$$  Prove that $$ f(z)=\frac{1}{2\pi i} \int_{\Gamma} \frac{f(\xi)}{z-\xi}d\xi,  $$ for all $z\in\mathbb{C}$ with $|z|&gt;2$.  I have no idea how to even start. Any hint will be appreciated.
76111
Suppose $a,b,c$ lie in the unit circle of the complex plane and satisfy $a+b+c = 0$. Then, $a,b,c$ form the vertices of an equilateral triangle. Try So, I want to show that $|b-a|=|c-a|=|b-c|$. We are given $|a| = |b| = |c| = 1$ and $a+b+c=0$. Since $|z| = z \overline{z} $, we have $$ (b-a)( \overline{b}-\overline{a} ) = (c-a)( \overline{c} - \overline{a} )$$ which simplifies to $$ |b| - b \overline{a} - a \overline{b} - |a| = |c| - c \overline{a} - a \overline{c} - |a| $$ Hence, $$ c \overline{a} + a \overline{c} - b \overline{a} - a \overline{b} = 0$$ $$ a( \overline{c} - \overline{b} ) + \overline{a} ( c-b) = 0 $$ Here I am stuck. Perhaps this is the wrong approach?
51856
Find the radius of the inscribed circle of a right triangle. The triangle's height is $\sqrt{6} + \sqrt{2}$ while the bisector of the right angle is 4.  This seemed like a generic similar triangles problem but it isn't that simple when I tried to solve it. Any help?  EDIT: Here is a picture of the problem
93231
Full question: Let $a$,$b$,$c$ be three positive integers such that $\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right) = 3$. Find those triples.  This is actually a national competition question in Vietnam (Violympic), which I have attended (and did poorly, but had a lot of fun).   I have understood almost every questions asked that day, but this one really makes my head pop, because I haven't learn much about integer solution equation and how to solve hard cases (like that one)  I have solved that a,b,c can't be all the same, because:  $$a = b = c$$ $$\implies(1+\frac{1}{a})(1+\frac{1}{b})(1+\frac{1}{c})=(1+\frac{1}{a})^3=3$$ $$⟺1+ \frac{1}{a}=\sqrt[3]{3}$$ $$⟺\frac{1}{a}=\sqrt[3]{3}-1$$ $$⟺a=b=c= \frac{1}{(\sqrt[3]{3}-1)}$$  And using , I found out that $(a,b,c) \in \{(1,3,8),(1,4,5),(2,2,3)\}$, but stuck at how to solve it.  Thank you in advance for checking out. I really appreciate your effort.
69364
Consider an isosceles triangle. Let r be the radius of the circumscribed circle and p be the radius of the inscribed circle. Prove that the distance d between the centers of these two circles is $$d = \sqrt{r(2p-r)}$$
116593
A circle has the same center as an ellipse and passes through the foci $F_1$ and $F_2$ of the ellipse,such that the two curves intersect in $4$ points.Let $P$ be any one of their point of intersection.If the major axis of the ellipse is $17$ and the area of the triangle $PF_1F_2$ is $30$,then find the distance between the foci.    Let the center of the ellipse and the circle be $(0,0)$ We are given $2a=$length of major axis$=17$. Let the coordinates of foci be $F_1(c,0)$ and $F_2(-c,0)$ We need to find $2c$. Area of $PF_1F_2=\frac{1}{2}\times 2c\times$perpendicular distance between $P$ and the axis of the major axis of the ellipse. I do not know how to solve it further.
115106
Let $f(z)=\frac{1}{(z-1)(z-2)}$ and $g(z)=\frac{z}{(z-1)(z-2)}$. Let $\gamma$ be the circle of radius 4 centered at the origin and traveled once in the counterclockwise direction. Let $A=\{ z \in \mathbb{C}: |z|&gt;3 \}$  1) I need to show there is no analytic function $G: A \to \mathbb{C}$ such that $G'(z)=g(z)$ for all $z$ in $A$.   2) I also need to show that there is an analytic function $F:A \to \mathbb{C}$ such that $F'(z)=f(z)$ for all $z$ in $A$.  Do I have to integrate to find such function?
54468
Consider the circle $$|z-6i|=3;$$ its centre is $(0,6)$ and its radius is $3$.  I want to find the greatest value of $\arg z$ achieved on this circle. My idea is that the tangents to the circle from the origin create right angles with the radius. Then use the Pythagorean Theorem and trigonometry to find the angle $\theta$ between the tangent and the imaginaery axis.  Then the formula for finding the two values of $\arg z$ in the form $p\pi$ where $-1\lt{p}\le1$ are: $$\frac{\pi-2\theta}{2}=argz$$ $$\frac{\pi-2\theta}{2}+2\theta=argz$$
60079
Question 1: Let $z$ and $w$ be complex numbers satisfying $|z| = 4$ and $|w| = 2$. Then enter in the numbers $$ |z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 $$ below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.  I plugged in $4$ and $2$ to each equation and got $36$, $64$, $4$, $4$, for question one. It was wrong.  Question 2: Let $z$ and $w$ be complex numbers satisfying $|z| = 5, |w| = 2,$ and $z\overline{w} = 6+8i.$ Then enter in the numbers $$ |z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2 $$ below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.  I don't know how to even start on question 2 because of the $z\overline{w}$.
114924
Why if the absolute value is the distance from $0$ on the number line, is $\lvert 3-\pi \rvert = \pi - 3$ and $\left\lvert\sqrt{2}-1\right\rvert= \sqrt{2}-1$