In a $\triangle ABC,$ If $\cot A+\cot B+\cot C =\sqrt{3},$ Then prove that $\triangle$ is equilateral.   $\bf{My\; Try::}$ Using Jensen's Inequality,  Let $f(x)=\cot x\;,$ Where $x\in (0,\pi),$ Then $f'(x) = -\csc^2 x$ and $f''(x) = 2\csc^2 x\cdot \cot x$  So we get $\displaystyle f''(x) = \frac{2\cos x}{\sin^3 x}&gt;0$ in $\displaystyle x\in (0,\pi)-\left\{\frac{\pi}{2}\right\}$  So $$\displaystyle\frac{\cot A+\cot B+\cot C}{3}\geq \cot\left(\frac{A+B+C}{3}\right)=\frac{1}{\sqrt{3}}$$  So we get $\cot A+\cot B+\cot C\geq \sqrt{3}$  But i did not understand how can we prove that $\triangle$ are equilateral  Thanks
If $\cot A+\cot B+\cot C=\sqrt3$ then prove that the triangle is equilateral triangle.   Trial: I can counter check that this is true as $\cot 60+\cot 60+\cot 60=3 \frac{1}{\sqrt3}=\sqrt3$.Here I also know that $A+B+C=180$. But I am unable to show it. Please help.
Suggested improvements to badges index page ()  Currently,  shows the possible badges, number of total recipients, and brief descriptions. However, I have a few suggestions (mostly minor changes) that could make it a bit nicer.   There is no way to see (on that page) which badges I've earned multiple times — they all appear with a green checkmark. It would be nice to show something like "x 4" like in the badges section of the user profile page. For example:      There is no way to see which badges can be awarded multiple times — you have to click on a specific badge to see yea or nay. It would be nice to have a small, simple icon next to badges that can be earned multiple times, similar to the checkmark showing which badges I've earned. The layout would be nicer if the number of recipients for each badge were in a separate column — since the badge images are different sizes, the overlap and ragged right alignment makes the numbers harder to read. An alternative view that groups badges by "achievement level" (gold/silver/bronze) would be nice, as would a view that groups progressive degrees of the same accomplishment. (e.g. Nice Answer &rarr; Good Answer &rarr; Great Answer OR Popular Question &rarr; Notable Question &rarr; Famous Question) python: restarting a loop i have:  for i in range(2,n):     if(something):        do something     else:        do something else        i = 2 **restart the loop   But that doesn't seem to work. Is there a way to restart that loop?   Thanks mysqli_fetch_assoc() expects parameter / Call to a member function bind_param() errors. How to get the actual mysql error and fix it? Arithmetic progression topology This is an example from Munkres's Topology (Example 4 in Section 22 titled &quot;The Quotient Topology&quot;, 2nd edition).  Example 4: Let $X$ be the closed unit ball $$\{ x \times y \mid x^2 + y^2 \le 1\}$$ in $\mathbb{R}^2$, and let $X^{\ast}$ be the partition of $X$ consisting of all the one-point sets $\{ x \times y \}$ for which $x^2 + y^2 &lt; 1$, along with the set $S^1= \{ x \times y \mid x^2 + y^2 = 1 \}$. Typical saturated open sets in $X$ are pictured by the shaded regions in the figure below. One can show that $X^{\ast}$ is homeomorphic with the subspace of $\mathbb{R}^3$ called the unit 2-sphere, defined by $$S^2 = \{ x \times y \times z \mid x^2 + y^2 + z^2 =1 \}.$$  I am confused about two points in the example.  Problem 1: About the saturated open sets in $X$. In the example, the typical ones are pictured by the shaded regions ($U, V$) in the figure. However, I am not sure whether the boundaries (visually, by picture) of $U$ and $V$ are included. Particularly, there are two boundaries for $U$. Are they both contained in $U$? And why? Problem 2: How to show that $X^{\ast}$ is homeomorphic with $S^2$? What is the mapping between them?  The following is my understanding:  Solution to problem 1: For $V$, the boundary is not included. For $U$, the outer boundary is included, while the inner one not. Is this solution right? Multiply cited reference in footnotes How does the bounty system work? It's possible to offer and award bounties of reputation on questions.   What is a bounty? How can I find questions that have a bounty attached? How do I start a bounty? When can I start a bounty? Why is the system requiring me to choose a bounty amount that's higher than the minimum? Can I use Markdown formatting in the bounty remarks? Can I edit the remark? How long is the bounty period? How do I award a bounty? Can I award a bounty to my own answer? Can I award a bounty to an old answer? If I offer a bounty on someone else's question, can I award the bounty to their answer? After awarding the bounty, can I remove it or move it to another answer at a later time? What happens if there's no answer after the bounty period? What happens if I feel my question is still unanswered? What is automatic awarding? Are bounty awards exempt from the 200 points/day reputation limit? How does Community Wiki mode affect bounties? How does user deletion or post dissociation affect bounties? Can I award a bounty to an answer from a deleted user or one where the author was dissociated? Can I offer a second bounty after the first one has expired? Can I raise my bounty? What happens if a bounty question is closed or deleted? Why can't I vote to close or migrate a bounty question? What happens if I delete my account while I have an active bounty? Do I keep an awarded bounty when I delete my answer? What happens if a question where I awarded a bounty later gets deleted or migrated? Can I cancel my bounty? Can I affect / divide / share part of my bounty to two (or more) users? How many outstanding bounties can a single user have?      See also:  in the Help Center Short story where rain only on one side of the house gives away fake world? In the mid-90s I read a short story about a guy freaking out at his wife and therapist because as they were leaving the house to go on a train trip to New York City, he went back inside to grab something and noticed that out the back window it was raining, while out the front door it was clear and sunny.  This confirmed his existing paranoia that everything in his world was fake, and everyone he knew was either a robot or an actor, participating in a massive effort to fool him and keep him from remembering who he really was.  He eventually calms down and agrees to go on the trip. The story ends      with a twist where the wife and therapist have a conversation something like, that was a close one, he almost remembered. Now strike the house set and assemble the New York set.   Can anyone identify this story? It was probably in a magazine or anthology, either sci-fi or speculative fiction. Possible to create new Application Handler? Blender's application handlers are listed , but is it possible to create a new one?  For example, I'd like an application handler which is a callback for every time I add a strip to the Video Sequence Editor. What are the details of "Objective-C Literals" mentioned in the Xcode 4.4 release notes? Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$ If the order of $G$ is $p^2$ then how do I show that $G$ is isomorphic to $\mathbb Z_{p^2}$ or $\mathbb Z_p\times\mathbb Z_p$. List comprehension with expensive computation Why has Earth's core not become solid? Is there a modal command for MakeLowercase? Analog to  question I've been wondering WHAT is to \lowercase as \bfseries is to \textbf? (... and for uppercase, or \MakeLowercase and MakeUppercase, etc., which also seem to be text-block commands as explained .)  Is there a "switch"/"declaration"/"modal command" (what is the correct term here?, is there a difference between them?) that would allow one to turn something into lowercase/uppercase in the same way that one would use \scshape for example, i.e., not as text-block command with the curly braces?  Or how to create such a command?  I couldn't find any clue on the forum. It is my understanding that I can disable my touchscreen by adding Option "Ignore" "on" to this file                                /usr/share/X11/xorg.conf.d/10-evdev.conf   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "on"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           EndSection   so it would look like this   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "on"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           Option "Ignore" "on"           EndSection   What would happen if I simplied changed MatchIsTouchscreen to "off"?   Section "InputClass"           Identifier "evdev touchscreen catchall"           MatchIsTouchscreen "off"           MatchDevicePath "/dev/input/event*"           Driver "evdev"           EndSection   Would this disable my touchscreen? Does registering [name]overflow.com violate Stack Overflow's trademark? Creating if-else condition with ArcGIS API for JavaScript combo boxes? Books for Condensed Matter after Ashcroft/Mermin What are some good condensed matter physics books that can fill the gap between Ashcroft &amp; Mermin and research papers? Suggestions for any specialized topics (such as superconductivity, CFT, topological insulators) are welcomed. Can I manually put an external hard disk to sleep? How to read negative radians in the interval? Say a function's domain is $\{-\pi/2, \pi/2\}$. How should I interpret this interval?  It starts from where? To where? In what direction?  And if it starts from $3\pi/2$, would the next one be $-5\pi/3$.(because it starts from negative, $-\pi/2$).