In any $∆ABC$, prove that: $$r^2+ r_1^2+ r_2^2 +r_3^2=16R^2-a^2-b^2-c^2$$ where $r$ is the inradius, $r_1,r_2,r_3$ are the exradii of the triangle respectively, and $R$ is the circumradius of the triangle. I tried to solve the question by replacing $r$ by $∆/s$ and $r_1$ by $∆/(s-a)$, and similarly for $r_2$ and $r_3$, but the answer is too long and I think there must be another short method.
From SL Loney Trigonometry:     Prove that   $$r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2$$   for a triangle with sides $a$, $b$, $c$, circumradius $R$, inradius $r$, and exradii $r_1$, $r_2$, $r_3$.   This one is a really dirty question.   I have tried a lot of stuff like $r=4R\times \Pi(\sin\frac{A}{2}$) and  $r_1=4R\times\sin\frac{A}{2}\cos\frac{B}2\cos\frac{C}2$ etc..  There are other ways like $-r+r_1+r_2+r_3=4R$,  but all these are way too cumbersome or clumsy.   Anything more intuitive?
An idiom to describe someone who thinks he/she is wiser than others Display top tags & percentiles from additional Stack Exchange sites on Careers Are the "top tags &amp; percentiles" from Stack Exchange sites other than Stack Overflow displayed in the Careers site?  If not can they be added?  If they are, then what are the requirements for them being displayed in the Careers site? When using divs when is it best to use a class vs id?   Is it best to use class, on say font variant or elements within the html? Then use id for the structure/containers?   This is something I've always been a little uncertain on, any help would  be great. Are the leftovers from making stock good for anything? Having made stock and strained it, can the meat and vegetables boiled up in the stock be used for anything, or should they go in the bin? How to determine function name from inside a function Helpful flags not getting updated after getting 'close question' privilege Possible Duplicate:          I recently got the ability to cast close votes. So, even if I flag a question as 'not constructive' or 'not a real question', it is still considered as a close vote. What is strange is that my 'helpful flag' count has stopped increasing after this change. Or at least it doesn't increase in cases where my flagging was automatically changed to a close vote and the post actually was closed. Is this how it is supposed to be?   I am guessing that once the post gets closed, the moderators are probably not bothered about marking flags as helpful. The only problem is that the Deputy badge will be much more difficult to get now. Earlier all the flags were getting converted to helpful, but now they just go in as close votes and don't show up on my profile page. General Relativity Equivalence Is Einsteins Equivalence theorem in General Relativity correct? It seems to me that it neglects the fact that gravitational acceleration depends upon separation distance squared, thus neglecting the effect of tidal forces.   For example, as I sit on earth, I experience the affect of earth's gravity; Although the acceleration on my head is slightly less than the acceleration acting on my feet. If I make the claim that my frame is equivalent to me being in a space ship traveling at g, doesn't that mean my whole body is accelerating at g uniformly? This is contrary to the previous statement though. Using headset controls like a smart phone Is there a way to simulate a numeric keypad on apple wireless keyboard in OSX? I'm tempted to get an Apple Wireless Keyboard but the lack of a number pad is a concern. I'm interested whether there is some form of software you can use where for example you hold some control keys down and you can use a set of keys like the following as a substitute numeric keypad:   7890 uiop jkl; m ./ Drop Tables with Bigger size We have a mysql database with a size of 2 TB, in which we have for to five big tables ranging from 110GB to 350GB. When we try to drop these tables the mysql goes to the defunct mode and hangs in that state for a long time. Is there a solution to drop these tables without mysql hanging. We have a machine which has a memory of 122GB and swap of 60 GB. We use an InnoDB Engine Hope the information suffices kindly help us with this issue What is the best time to ask questions? I have a SQL Server 2005 database using the full recovery model. If I execute the following commands:  ALTER DATABASE test SET RECOVERY SIMPLE; ALTER DATABASE test SET RECOVERY FULL;   or   BACKUP LOG test WITH  truncate_only;   Then try to run a log backup:  BACKUP LOG test TO  DISK = 'backupfile.bak'   I receive the error message:     BACKUP LOG cannot be performed because there is no current database backup.   Question:     How do I check that I will not be able to take a log backup, without running a backup log command? For a command, if using - as an argument in place of a file name will mean STDIN or STDOUT.   But in this example, it creates a file with the name -:  echo hello &gt; -   How can I make - in this example mean STDOUT? Conversely, how can I make - mean a file named - in examples such as:  cat - Why cant we replace diode connected BJTs by simple diodes in a current mirror circuit? How fast does gravity propagate? When running apt-get -y install &lt;packages ...&gt; on Ubuntu 10.04 I would like apt-get (or aptitude if that makes it easier) to not prompt me when installing additional dependencies (behavior of -y as I understand it) and but not prompt me about overwriting configuration files, instead assume to keep the existing ones always (which is usually the default). Unfortunately --trivial-only seems to be the inverse of -y and not affect the prompt that is shown, according to the man page.  In particular packages auch as samba, nullmailer, localepurge and lighttpd have forced me to interact with the terminal, even though the whole procedure was scripted and meant to be non-interactive. QGIS color polygon according to points I have two layers. Layer 1 contains points (x,y) and layer 2 contains one polygon. All points in layer 1 are positioned within the polygon of layer 2. The points in layer 1 contain an attribute with information on the topography/elevation at the specific point. Can I color the polygon in layer 2 based on attributes from the points in layer 1 in order to create an elevation map? I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup of order $n$"). This is an exercise from a book, and it is in the section on finite abelian groups, so I know I have to use the fundamental theorem of finite abelian groups. I have come up with a proof, but it seems a bit messy, and I'm not entirely sure if it's correct. It is given below.     Let the order of $G$ be $m$ = $p_1^{\alpha_1} \ldots p_k^{\alpha_k}$. It is known that $G$ is a direct product of $p$-groups, say:      $$G = G_1 \times \ldots \times G_k$$      where each $G_i$ is a $p_i$-group. By the fundamental theorem of finite abelian groups, each $G_i$ is isomorphic to a direct product of cyclic groups of the form      $$\mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}},$$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$. Now if $n$ divides $m$, then we must have      $$n = p_1^{\gamma_1} \ldots p_k^{\gamma_k}$$      for some $\gamma_1, \ldots, \gamma_k$ with $0 \leq \gamma_i \leq \alpha_i$.      Claim: Each $G_i$ has a subgroup of order $p_i^{\gamma_i}$      Proof: As above, we have that       $$ G_i \cong \mathbb{Z}_{{p_i}^{\beta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\beta_l}} $$      where $\beta_1, \ldots, \beta_l$ are positive integers such that $\sum_{j=1}^l \beta_j = \alpha_i$.      Now since $0 \leq \gamma_i \leq \alpha_i$, we can find   $l$ numbers $\delta_1, \ldots , \delta_l$ such that $\gamma_i = \sum_{j=1}^l \delta_j$, and $0 \leq \delta_j \leq \beta_j$. (This choice of numbers is not necessarily unique).      Then $p_i^{\delta_j} | p_i^{\beta_j}$ for each $j = 1, \ldots , l$. Hence, for each factor $\mathbb{Z}_{{p_i}^{\beta_j}}$, there exists a subgroup of order $p_i^{\delta_j}$, namely $\mathbb{Z}_{{p_i}^{\delta_j}}$ (using the fact that the converse of Lagrange's theorem is true for finite cyclic groups). Taking the direct product of each of these subgroups, we get a new subgroup $G_i'$ of $G_i$:      $$G_i' \cong \mathbb{Z}_{{p_i}^{\delta_1}} \times \ldots \times \mathbb{Z}_{{p_i}^{\delta_l}}$$      The order of this subgroup is $p_i^{\delta_1} \times \ldots \times p_i^{\delta_l} = p_i^{\delta_1 + \ldots + \delta_l} = p_i^{\gamma_i} $. So we have found a subgroup of $G_i$ of order $p_i^{\gamma_i}$, as required.      So each factor $G_i$ in the product $G = G_1 \times \ldots \times G_k$ has a subgroup $G_i'$ of order $p_i^{\gamma_i}$.       Therefore, $G$ has a subgroup   $$G_1' \times G_2' \times \ldots \times G_k'$$       of order $p_1^{\gamma_i}...p_k^{\gamma_k} = n$, which completes the proof.   I have two questions about this: firstly, does this proof seem to work? Secondly, is there a way to make the proof more concise (e.g. a way to prove the statement without using all these indices)? How to avoid the wrinkle at the poles of the UV sphere? SElinux: allow httpd to connect to a specific port My system is running CentOS 6.4 with apache2.2.15. SElinux is enforcing and I'm trying to connect to a local instance of redis through my python/wsgi app. I get Error 13, Permission denied. I could fix this via the command:  setsebool -P httpd_can_network_connect  However, I don't exactly want httpd to be able to connect to all tcp ports. How can I specify which ports/networks httpd is allowed to connect to? If I could make a module to allow httpd to connect to port 6379 ( redis ) or any tcp on 127.0.0.1, that would be preferable. Not sure why my paranoia is so strong on this, but hey...  Anyone know?