Prove that $r^2+ r_1^2+ r_2^2 +r_3^2=16R^2-a^2-b^2-c^2$
Prove $r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2$ for a triangle with sides $a$, $b$, $c$, circumradius $R$, inradius $r$, exradii $r_1$, $r_2$, $r_3$
How to return only the Date from a SQL Server DateTime datatype Is Firefox 57 multithreaded? According to  on MDN, Firefox 57 finally runs things in different processes.  So is Firefox 57 multithreaded?  In particular, how many cores does it use at most? Say I open 20 tabs at once on a quad-core machine, will Firefox 57 launch as many threads, or at least as many as needed to max out my CPU?  (Chromium for instance spawns a number of sub-processes and very easily uses all the resources available. Firefox up to 56 in GNU/Linux is rarely seen going over 135 % of one core however many tabs I open.) Different seed gives different LASSO output When I try to connect to servers or join up on lan worlds in Minecraft it says "Failed to Login : Invalid session (Try restarting your game)". Multiplayer used to work fine but now keeps coming up with this error. Can anybody help? I'm trying to solve the following equation:  $$x^3-y^3=xy+61$$  I got:  $$(x-y)(x^2+xy+y^2)=xy+61$$  But I can't go any further. I am looking for a solution in integers.  I need some hints to proceed, please! How to use San Francisco font on Windows? How does LSTM prevent the vanishing gradient problem? LSTM was invented specifically to avoid the vanishing gradient problem. It is supposed to do that with the Constant Error Carousel (CEC), which on the diagram below (from ) correspond to the loop around cell.  (source: ) And I understand that that part can be seen as a sort of identity function, so the derivative is one and the gradient stays constant. What I don't understand is how it does not vanish due to the other activation functions ? The input, output and forget gates use a sigmoid, which derivative is at most 0.25, and g and h were traditionally tanh. How does backpropagating through those not make the gradient vanish ? Why screen resolution remains 640x480 when I close a game? Significance of assumption in competition inequality questions Refer to the problem below (IMO 2009 Shortlist)     Let $a, b, c$ be positive real numbers such that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = a + b+ c$$   Prove that    $$\frac{1}{(2a+b+c)^2} + \frac{1}{(2a+b+c)^2} + \frac{1}{(2a+b+c)^2} \leqslant \frac{3}{16} $$    In the solution given by the  (Pg 16), it states that      Without loss of generality, we choose $$a +b+c = 1$$.    Thus, the problem becomes    $$ \frac{1}{(1 + a)^2} + \frac{1}{(1 + b)^2} +\frac{1}{(1 + b)^2} \leqslant \frac{3}{16}(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$$    Applying Jensen’s inequality to the function $$f(x) = \frac{x}{ (1+ x)^2}$$   ,which is concave for $0 ≤ x ≤ 2$   and increasing for $0 ≤ x ≤ 1$, we obtain   $$α \frac{a}{(1 + a)^2} + β \frac{b}{(1 + b)^2} + γ \frac{c}{(1 + c)^2} \leqslant (α + β + γ) \frac{A}{(1 + A)^2}$$   , where $A =\frac{αa + βb + γc}{α + β + γ}.$   Choosing $α = \frac{1}{a} , β =\frac{1}{b},$ and  $γ = \frac{1}{c}$   , we can apply the harmonic-arithmetic-mean inequality   $$A =\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} ≤ \frac{a + b + c }{3} = \frac{1}{3} &lt; 1$$   Finally we prove:   $$ \frac{1}{(1 + a)^2} + \frac{1}{(1 + b)^2} +\frac{1}{(1 + b)^2} \leqslant ( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \frac{A}{(1 + A)^2} \leqslant ( \frac{1}{a} + \frac{1}{b} + \frac{1}{c})\frac{A}{(1 + \frac{1}{3})^2} = \frac{3}{16}(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) $$   However,  (Pg 4) differs by having $a +b+c=3$ and setting $f(x) = \frac{1}{16x} - \frac{1}{(x+3)^2}$ and allowing Jensen to resolve the rest.   The questions are as follows;   How do you choose before hand which $a + b+ c =$ to choose?  Why particularly does the assumption differ between the 2 solutions? How does one know before hand the choosing of $α = \frac{1}{a} , β =\frac{1}{b},$ and  $γ = \frac{1}{c}$?  Where and how did $A =\frac{αa + βb + γc}{α + β + γ}$ come from?   Any help would be much appreciated. Prove $[\sin x]' = \cos x$ without using $\lim\limits_{x\to 0}\frac{\sin x}{x} = 1$ I came across this question:   From the comments, Joren said:     L'Hopital Rule is easiest: $\displaystyle\lim_{x\to 0}\sin x = 0$ and $\displaystyle\lim_{x\to 0} = 0$, so $\displaystyle\lim_{x\to 0}\frac{\sin x}{x} = \lim_{x\to 0}\frac{\cos x}{1} = 1$.   Which Ilya readly answered:     I'm extremely curious how will you prove then that $[\sin x]' = \cos x$   My question:  is there a way of proving that $[\sin x]' = \cos x$ without using the limit $\displaystyle\lim_{x\to 0}\frac{\sin x}{x} = 1$. Also, without using anything else $E$ such that, the proof of $E$ uses the limit or $[\sin x]' = \cos x$.    All I want is to be able to use L'Hopital in $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}$. And for this, $[\sin x]'$ has to be evaluated first.    Alright... the definition that some requested.   Def of sine and cosine: Have a unit circumference in the center of cartesian coordinates. Take a dot that belongs to the circumference. Your dot is $(x, y)$. It relates to the angle this way: $(\cos\theta, \sin\theta)$, such that if $\theta = 0$ then your dot is $(1, 0)$.  Basically, its a geometrical one. Feel free to use trigonometric identities as you want. They are all provable from geometry. Norm of adjoint operator in Hilbert space Is it possible to trade CFD without leverage? I'm new to trading. Based on what I read, trading CFD involves high risk because of leverage.   Is it possible to trade CFD stocks without leverage (in Etoro for example)?  If this is possible, then I would lose/gain only the percentage by which the stock goes down/up.  Thank you How do I redirect in a class sub-classing the EventSubscriber class? 'Sorry' as a replacement for 'Excuse me'? Is it ok to say 'sorry' where you otherwise would use 'excuse me'?  Example: "Sorry, but could you show me the way to the next train station?" I have upgraded from Ubuntu 16.04 LTS to Ubuntu 18.04 LTS. I was surprised to see that the context menu entry "Make Link" (keyboard shortcut Ctrl+M) in Nautilus doesn't seem to exist anymore. It used to appear upon right mouse click on a file or folder and allowed the user to create a symbolic link. How do I identify SMD components? (or how do I identify any component) How do I identify the markings on an SMT component and match it up with a part number so I can be a good designer and actually look at a datasheet (and read the whole thing)? Or identify a part to replace an unknown part on a PCB? I'm writing a math paper in LaTeX and for math formulas I'm just using $$math formula$$. Is is possible to numerate formulas? Or do I have to you use some specific commands such as align? Force ssh to not print "Remote host identification has changed" warning Is there a way to avoid ssh printing warning messages like this?   @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ @    WARNING: REMOTE HOST IDENTIFICATION HAS CHANGED!     @ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@   Although the remote host identity has changed but I know it is fine and  just want to get rid of this warning. How to pronounce '-ing' followed by a vowel I'm getting into English recently and I'm a little confused by the way people pronounce a word that starts in a vowel right after a word ending in -ing.  For example:      You have to bring it up now?   I don't know if I should pronounce it as:   bring /ŋit/ brin' /nit/ bring /git/   Another example would be "I'm coming out".  Likewise, what about the ending -ang? For example:      We can just hang out and have a good time.    And to round it all up, what about a combination of the two? For example,      He got sick of hanging around waiting for you and went home.   I've been asking myself these questions for weeks. Limiting number of lines in textarea I'm looking for a javascript that can limit the number of lines (by line I mean some text ended by user pressing enter on the keyboard) the user is able to enter in textarea. I've found some solutions but they simply don't work or behave really weird. The best solution would be a jquery plugin that can do the work - something like , but it should be able to limit text line count not character count.