Prelude  While studying trigonometry, I came across this very interesting problem. It wasn't very difficult to solve, however it's result was quite interesting. I have given the solution below. Try to solve it yourself before looking.    Problem  The diagram below shows three equal squares, with angels $\alpha$, $\beta$, $\gamma$ as marked. Prove that $ \alpha + \beta = \gamma $.      Hint  If you are not in the midst of a trigonometry course, chances are you will need to see this hint, as it is an uncommon identity.  Hover below to see a hint.      $ \arctan a + \arctan b = \arctan{\frac{a + b}{1 - ab}} $   This identity was solved earlier in the book I found this problem in (see Postscript). Try to prove this identity for yourself, too!    Postscript  This problem came from the textbook: "Trigonometry" by I. M. Gelfand.
How can I show that $x+y=z$ in the figure without using trigonometry? I have tried to solve it with analytic geometry, but it doesn't work out for me.
I just asked my first question on stackoverflow and I'm really impressed that my question seems to be answered sufficiently already! See:    The issue I have is that the last two answers combined would provide the most complete answer. What's the proper way to go about combining them, marking the question answered, and giving credit where credit is due?  Thanks big time guys for saving us all from experts exchange hell!  Edit: Wow, 5 responses in 5 minutes. I think what makes the most sense is that I create my own answer out of the two. I don't think the second part of my question was answered though.  What is the proper way to give credit to the 2 answers that I'm combining? Is an upvote for each sufficient? How soon should I "vote to close"? Should we avoid negative words when writing error messages? How do they express the time, in American and British English? How can I enable shell-escape? I need to enable shell escape for a package that I want to use. What is shell escape exactly, and how do I enable it in my editor or compiling toolchain? Links to answers for different editors/toolchains, in alphabetical order:   AUCTeX (Emacs) ,         Note: this is a FAQ post. Related Meta question: . Please add an answer for your own editor/toolchain if it is missing. Thank you! When adding an answer, please indicate if the setting is permanent or per-document, and/or describe how you can add a button, shortcut, menu item, etc. to make it easy to switch between shell-escape and regular compilation. What to do when visa admission expires during lockdown? I was admitted to the US as a visitor for a period which expires soon.  Until tonight, I had solid plans to return to Europe soon, prior to the expiration of the visa entry, but that is no longer an option.  I do not wish to overstay or do anything that would cause problems with future admissions or visa applications.  However, I can't go back to Europe because of new travel restrictions put in place by the US government (/resulting flight cancellations).    My passport still has years of validity left.  What is the best course of action? Is there any official guidance on this subject?  I am a little concerned that contacting officials to explain the situation would just lead me to be flagged for prompt enforcement action, but I don't want future problems.    Related:  and .    Update: OP now planning an escape to Canada.  The question still stands, though. A problem of missing energy when charging a second capacitor A capacitor is charged. It is then connected to an identical uncharged capacitor using superconducting wires. Each capacitor has 1/2 the charge as the original, so 1/4 the energy - so we only have 1/2 the energy we started with. What happened? my first thoughts were that the difference in energy is due to heat produced in the wire. It may be heat, or it may be that this is needed to keep equilibrium. Why is $a^{-1}$ mod $p$ equal to $a^{p-2}$ mod $p$ in modular arithmetic? Reputation items in the achievements dropdown don't add up Let $X$ be topological space and $A\subseteq X$.   If $E\subseteq X$ is connected, $E\cap A\neq\emptyset$ and $E\cap(X\setminus A)\neq\emptyset$ then $E\cap\partial A\neq\emptyset$.  Assume that $E\cap\partial A=\emptyset$.  If $E\cap\partial A=\varnothing$ then $E=(E\cap\text{int}(A))\cup(E\cap\text{ext}(A))$, i.e. a union of disjoint sets open in $E$.   Also the sets are nonempty because: $E\cap\text{int}(A)=E\cap A\neq\varnothing$ and $E\cap\text{ext}(A)=E\cap A^c\neq\varnothing$.  This contradicts that $E$ is connected.  Fixed. Should I be offered the option to upgrade to 18.04.1? Filtered Questions Asked By Accepted Answer/ No Accepted Answer I think it's helpful if we can go to a page and view all our own questions that   Does not have an accepted answer, and/or Does not have an answer yet.   The reason I asked for them is because, I sometimes need to review through all the questions, and pick those that don't have proper answers so that I can perform further operations on them ( such as offering a bounty, or accept an existing answer). The Quotation Mark character in TeXworks editor has vanished Now, my version of TeXworks is 0.4.5r.1280 and I can't have the " character on the keyboard.    I have ``  or '' instead. Why?  And It's the same with Alt+034. I'm very disappointed with that. Google Maps Location History is showing an incorrect location I have a major problem that I hope someone can help with. On my Google Maps feature under my Location History it shows a location that I was never at at any point, but when my wife happened to see that location she doesn't believe me that the GPS could be wrong and pinpoint that location if I was never there.  I was first wondering myself how could that could be possible if I was never there or my phone was never at that location that was 35 miles away from me and second is there any way I could prove the GPS being wrong? Can I contact Google Maps and can they give an official printout of that certain day and every location I was? I really need to prove my honesty to her and hope there is a way. How can I display a short chapter name in the header and a long chapter name in the ToC? I am writing a book and one of my chapters has a very long title. As it can't be displayed correctly on the top of even pages, I provide a "short title name" via the   \chapter[short title]{long title}   command. But then, "short title" appears in the table of contents instead of the original/true/long one.  How can I have the "short title" in the header of even pages, and the "long title" in the ToC and in the "Title header"?  For reference, here is the code I use:  \documentclass[11pt,a4paper]{book} \begin{document} \tableofcontents  \chapter[Short title]{Very vey very very very very very very long title     i can't display in the header} Lorem ipsum dolor sit amet, consectetur adipiscing elit. \end{document} How to create an annuity timeline on LaTeX I was wondering, how can I create a timeline exactly like this one:    on LaTeX?  I am a Finance student, and write all of my notes using LaTeX. These timelines, are giving me a headache. Since we use a lot of different timelines because of annuities. I want to learn how to create one basic timeline, so that I can then change anything I want. But basically, I want something like this, with the arrows and everything.  Thank you so much guys! Can words like "what" be the subject of a sentence? In a question like "Who hears a noise?",  is the subject of the sentence who?  I can think of a few tests for subjects like: "the subject is the phrase that inverts with the auxiliary to form a question". But this is a question and there is no auxiliary. We could say that's because it's a subject, but that would be just to presuppose that it's a subject in the first place.  I can think of tests like: "the form of the verb will agree with the number of the subject", but on the other hand who does not seem to have  fixed grammatical number, and it sometimes seems like who might actually be agreeing with noun phrases later in the sentence:   Who are the most prolific writers of our age? Who is the most prolific writer of our age?   There are also tests like "the subject is usually the first noun phrase in the sentence". But of course if we have a question like:   Who have you bitten?   ... you, not who seems to be the subject. The first noun phrase test is not very good.  I also know that some theories of grammar say that there is a gap in wh- questions that the wh- word is extracted from. So can this gap for instance be the real object of a question? If so, is [See community wiki post in the linked to question]  How can we show whether who is really the subject of my original example question? And how about other wh- words like what? Can they (also) function as subjects? What is wrong with "Where should this car be parked?"? Loading .sql files from within PHP I am often logged in several SSH sessions at once. To logout from several sessions, I press CTRL+d, until I am back on my local machine. However, I occasionally press it once too many, and my terminal exits. Is there a way to make CTRL+d unable to close my terminal ? I am using terminator as my terminal emulator.