Prove that distance between orthocenter and incenter is $\sqrt{2r^2-4R^2\cos A\ cos B\cos C}$  My attempt is as follows:-    In the figure H denotes the orthocenter and I denotes incenter.  $IM=r$, $BH=2R\cos B$  In $\triangle IBM$  $$\dfrac{IB}{IM}=\mathrm{cosec} \frac{B}{2}$$ $$IB=r\mathrm{cosec} \frac{B}{2}$$ $$IB=4R\sin \frac{A}{2}\sin \frac{C}{2}$$  $\angle HBI=90-C-\dfrac{B}{2}$  Applying cosine rule in $\triangle IBH$  $$\cos \left(90-C-\dfrac{B}{2}\right)=\dfrac{BH^2+BI^2-HI^2}{2BH\cdot BI}$$  $$2\sin\left(C+\dfrac{B}{2}\right)\cdot2R\cos B\cdot4R\sin\frac{A}{2}\sin\frac{C}{2}=4R^2\cos^2B+16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-HI^2$$  $$HI^2=4R^2\cos^2B+16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-16R^2\sin\left(C+\dfrac{B}{2}\right)\cdot\cos B\cdot\sin\frac{A}{2}\sin\frac{C}{2}$$  $$HI^2=4R^2\cos^2B+16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-16R^2\sin\left(\dfrac{2C+B}{2}\right)\cdot\cos B\cdot\sin\frac{A}{2}\sin\frac{C}{2}$$  $$HI^2=4R^2\cos^2B+16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-16R^2\sin\left(\dfrac{2C+\pi-A-C}{2}\right)\cdot\cos B\cdot\sin\frac{A}{2}\sin\frac{C}{2}$$  $$HI^2=4R^2\cos^2B+16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-16R^2\sin\left(\dfrac{\pi}{2}-\dfrac{A}{2}+\dfrac{C}{2}\right)\cdot\cos B\cdot\sin\frac{A}{2}\sin\frac{C}{2}$$  $$HI^2=4R^2\cos^2B+16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-16R^2\cos\left(\dfrac{A}{2}-\dfrac{C}{2}\right)\cdot\cos B\cdot\sin\frac{A}{2}\sin\frac{C}{2}$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-16R^2\cos\left(\dfrac{A}{2}-\dfrac{C}{2}\right)\cdot\cos B\cdot\sin\frac{A}{2}\sin\frac{C}{2}+4R^2\cos^2B$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\left(4\cos\left(\dfrac{A}{2}-\dfrac{C}{2}\right)\cdot\sin\frac{A}{2}\sin\frac{C}{2}-\cos B\right)$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\left(4\sin\frac{A}{2}\sin\frac{C}{2}\left(\cos\frac{A}{2}\cos\frac{C}{2}+\sin\frac{A}{2}\sin\frac{C}{2}\right)-\cos B\right)$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\left(4\sin\frac{A}{2}\sin\frac{C}{2}\cos\frac{A}{2}\cos\frac{C}{2}+4\sin^2\frac{A}{2}\sin^2\frac{C}{2}-\cos B\right)$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\left(\sin A\sin C+4\sin^2\frac{A}{2}\sin^2\frac{C}{2}-\cos B\right)$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\left(\sin A\sin C+(1-\cos A)(1-\cos C)-\cos B\right)$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\left(\sin A\sin C+1-\cos C-\cos A+\cos A\cos C-\cos B\right)$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\left(1-\cos C-\cos A+\cos A\cos C+\sin A\sin C-\cos B\right)$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\left(1-\cos C-\cos A+\cos A\cos C+\sin A\sin C-\cos B\right)$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4R^2\cos B\cos A\cos C-4R^2\cos B\left(1-\cos C-\cos A+\sin A\sin C-\cos B\right)$$  $$HI^2=4R^2\left(4\sin^2\frac{A}{2}\sin^2\frac{C}{2}-\cos B\left(1-\cos C-\cos A+\sin A\sin C-\cos B\right)\right)-4R^2\cos B\cos A\cos C$$  $$HI^2=4R^2\left(4\sin^2\frac{A}{2}\sin^2\frac{C}{2}-\cos B\left(1-\cos C-\cos A+\sin A\sin C-\cos (\pi-A-C)\right)\right)-4R^2\cos B\cos A\cos C$$  $$HI^2=4R^2\left(4\sin^2\frac{A}{2}\sin^2\frac{C}{2}-\cos B\left(1-\cos C-\cos A+\sin A\sin C+\cos A\cos C-\sin A\sin C\right)\right)-4R^2\cos B\cos A\cos C$$  $$HI^2=4R^2\left(4\sin^2\frac{A}{2}\sin^2\frac{C}{2}-\cos B\left(1-\cos C-\cos A+\cos A\cos C\right)\right)-4R^2\cos B\cos A\cos C$$  $$HI^2=4R^2\left(4\sin^2\frac{A}{2}\sin^2\frac{C}{2}-4\cos B\sin^2 \frac{A}{2}\sin^2\frac{C}{2}\right)-4R^2\cos B\cos A\cos C$$  $$HI^2=16R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}\left(1-\cos B\right)-4R^2\cos B\cos A\cos C$$  $$HI^2=32R^2\sin^2\frac{A}{2}\sin^2\frac{C}{2}\sin^2\frac{B}{2}-4R^2\cos B\cos A\cos C$$  $$HI^2=2r^2-4R^2\cos B\cos A\cos C$$ $$HI=\sqrt{2r^2-4R^2\cos B\cos A\cos C}$$  This proof got very long, any simple way to prove this?
I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius. I considered $\triangle API$ ($P$ is orthocentre and $I$ is incentre). I could find  $AP=2R\cos A$, $AI=4R\sin\frac{B}{2}\sin\frac{C}{2} $ and $\angle PAI=\angle \frac{B-C}{2}$. So applying cosine rule I got $$PI^2=4R^2+16R^2\sin^2\frac{B}{2}\sin^2\frac{C}{2} -16R^2\cos A\sin\frac{B}{2}\sin\frac{C}{2}\Bigg(\cos\frac{B}{2}\cos\frac{C}{2}+\sin\frac{B}{2}\sin\frac{C}{2}\Bigg)$$ How to proceed form here ?
I found the below in  web-site:     We use much in questions and negative clauses to talk about degrees of something.      We don’t use much in affirmative clauses.   Also, it the same web-site, I found the following:     She runs much faster than he does.    also I noticed that some web-sites use much with affirmative clauses:     May we all be having this much fun and getting this much exercise when we're his age!    Is it Okey to use much in affirmative clauses? If yes, when can we use it? and what is the rules?  Update:  Here is another example on how often the word "much" is used  in informal style:     After much consideration I will NOT be performing at the Inauguration with my improv group Capitol Hill-arious SSH: How to disable weak ciphers? Security team of my organization told us to disable weak ciphers due to they issue weak keys.    arcfour   arcfour128   arcfour256   But I tried looking for these ciphers in ssh_config and sshd_config file but found them commented.   grep arcfour * ssh_config:#   Ciphers aes128-ctr,aes192-ctr,aes256-ctr,arcfour256,arcfour128,aes128-cbc,3des-cbc   Where else I should check to disable these ciphers from SSH? (semi)-Automatic pluralization around a citation How to iterate over arguments in a Bash script I have a complex command that I'd like to make a shell/bash script of.  I can write it in terms of $1 easily:  foo $1 args -o $1.ext   I want to be able to pass multiple input names to the script. What's the right way to do it?    And, of course, I want to handle filenames with spaces in them. Do database engines update when the updated value is the same? Average time alloted for manuscript review across disciplines I am asking this question because I've always had the impression that manuscripts submitted to a journal in my field (theoretical linguistics) take an insanely long time to get published, and sometimes I get the impression that this is because, most of the time, nobody other than the author cares about getting things done within a reasonable amount of time. Some examples from personal experience follow.   For the last paper I submitted before getting my PhD, I had to wait nine months from the submission date to get reviews back from the referees. Last week I got a review request from a journal I had never reviewed for previously. The review deadline is March 15 (almost four months). In early October, I submitted a manuscript to a journal that I know asks reviewers to return reviews within 6 weeks. I checked the status of the manuscript online and it still says "with editor". Earlier this year, I was about to start writing a review when my wife went into labor. By the time we came back from the hospital, I had forgotten about the review, so I missed the deadline. The editor didn't contact me to ask about the review until two months after the deadline. Late in 2013, I was asked to write a survey chapter for a handbook. The expected publication date of the handbook is summer 2016.   All of this is not me being unlucky with editors and referees. In private conversations through the years, many colleagues have confirmed that it is virtually unheard of to get reviews back within 3 months of submission; 5 or 6 months seems to be a good average, but in some cases it can take longer (see my first point above). In general, everybody seems to have accepted that the submission-to-printed-journal process is going to take a bare minimum of one year, and that's assuming that the manuscript is accepted without revisions. If revisions are required, two years is not out of the question. Also, it's not like our papers are ridiculously long. I've reviewed about 100 papers so far, and the longest one was about 50 single-space pages; 30-35 single-spaced pages is closer to average.  I remember that, when I was a postdoc, my astrophysics housemate mentioned that in his field the entire submission-to-printed-journal process took only three or four months. I really can't see a good reason why theoretical linguistics shouldn't operate on similar timeframes, other than the fact that everybody seems to have grown used to things taking way longer than they really need to. What are the wait times in other fields? How do you manage to get things done quickly? Is it possible to get the progress of an XMLHttpRequest (bytes uploaded, bytes downloaded)?   This would be useful to show a progress bar when the user is uploading a large file. The standard API doesn't seem to support it, but maybe there's some non-standard extension in any of the browsers out there? It seems like a pretty obvious feature to have after all, since the client knows how many bytes were uploaded/downloaded.  note: I'm aware of the "poll the server for progress" alternative (it's what I'm doing right now). the main problem with this (other than the complicated server-side code) is that typically, while uploading a big file, the user's connection is completely hosed, because most ISPs offer poor upstream. So making extra requests is not as responsive as I'd hoped. I was hoping there'd be a way (maybe non-standard) to get this information, which the browser has at all times. Generic constraints, where T : struct and where T : class How do I remove login credentials (emails or OpenIDs) from my account?     For more information, see "" in the . How are percentiles calculated in Careers 2.0? I noticed that a percentile is placed at the top of my resume in Careers 2.0.    How are these percentiles calculated? Is there a certain threshold you need to be over in order for it to be displayed? Out of five needed members for my doctoral thesis committee, four are fixed. For the fifth member, I approached Mr.A. He has no real connection to my research or topic but I contacted him because he was just as good a fit as anybody else in the department to be the fifth member. I have never worked with him or taken any classes but we did talk a lot in the first few years where he would just ask about my progress by the copy machine, etc. We don't even have a "rapport". The relationship is absolutely neutral, nothing positive/friendly, and nothing negative. I asked him to be on my committee and he said yes. I said I'll get back to you with dates, etc. He is a senior tenured professor.  Then about a month later, I was looking at the CV of some of the new hired (junior) faculty in the department Mr.B (this is his first year) and I see that we have a collaborator in common. Mr.B has published multiple papers with him and Mr.B is working with the same code and the code managing team that we are working with. He is definitely a good person to have on my committee. I finally see Mr.B after a few more months (because of busy schedule/travel/etc.) and he agrees to be on my committee.  The question is, how can I tell Mr.A, as tactfully as possible, after about four months of being quiet that he is basically rejected from my committee because I didn't know about Mr.B but now that I know, Mr.B is a much better match than he is? That I don't need him but I do appreciate his offer? I don't want to antagonize Mr.A or burn any bridges. Should this be in email or in person? Should I explain or just keep it terse? Should I keep it vague that I don't need him and thanks for his time? Or should I tell him about Mr.B and why he is a better match?  FYI, this was before anything official/paperwork was done so there is no problem with that. I asked a  recently, looking for examples of an online tool. Per the , I answered my own question with a few examples that I had already found. I did this so that these examples would be included in the answers, along with the rest.  The question has received only down votes and insults saying that this shouldn't be done.   Should I not answer my own questions? Intuition behind the Definition of Conditional Probability (for 2 Events) What is some intuitive insight regarding the conditional probability definition: $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$ ? I am looking for an intuitive motivation. My textbook merely gives a definition, but no true development of that definition. Hopefully that's not too much to ask. I was trying to fix my audio problems using suggestions from . One of the comments suggested I execute this and only this:     killall pulseaudio   So I did, and I restarted my computer just in case, but all it did was remove my soundcard and I still couldn't hear anything. Now I don't know how to get my soundcard back.  Please note that I am extremely inexperienced when it comes to Ubuntu. Crop jpeg into circular tikz node \documentclass[demo]{article} \usepackage{graphicx} \usepackage{tikz} \begin{document} \begin{tikzpicture} \node[circle,draw,inner sep=2cm]{\includegraphics{myimage.jpg}}; \end{tikzpicture} \end{document}   How do I modify the above code so that the included image is adapted to the circle shape and cropped of the exceeding parts?  I want to specify a shape (a circle) and then include an image respecting the given dimension and cropped. Let $f: M \to N$ be a mono in $\mathsf{Mon}$. I want to prove that the underlying function $U(f): U(M) \to U(N)$ is injective.   How can this be done ? Thanks in advance. Install Linux from Linux Where can I see the source code of the Sun JDK? Neural network - meaning of weights Does this constitute a valid proof that $\frac{x^2}{1+x^4} \leq \frac{1}{2}$?