Consider this nice little problem: if $ABCD$ is a quadrilateral inscribed in a unit square, then $$2\leq AB^2+BC^2+CD^2+DA^2\leq4$$ (Evidently this is . I also found it in another nice collection of problems, .)  Here's the straightforward algebraic proof I used: let $x_1$, $x_2$, $x_3$, $x_4$ denote legs of the right triangles with hypotenuses $AB$, $BC$, $CD$, $DA$, with no two of the $x_i$ lying on the same side of the square. Then the Pythagorean theorem gives $$AB^2+BC^2+CD^2+DA^2=\sum_{i=1}^4 x_i^2+(1-x_i)^2=\sum 1-2x_i(1-x_i)=4-2\sum x_i(1-x_i)$$ But $$0\leq x_i(1-x_i)\leq\frac{1}{4}$$ and the claim follows. (The lower bound of 2 is achieved when the vertices of $ABCD$ are the midpoints of the sides of the square, and the upper bound of 4 is achieved when $ABCD$ is the square itself.)  I'm curious about other approaches to the problem. Perhaps the result falls out of a more high-powered theorem in geometry? Or does the problem essentially boil down to a nice application of clever quadratic algebra?
Let $S$  be a square of unit area. Consider any quadrilateral whose $4$ vertices lie on each side square $S$. Let the length of the sides of this quadrilateral be $a,b,c,d$. Then prove that $$2 \leq a^2+b^2+c^2+d^2 \leq 4$$     This problem appeared in IIT JEE $1997$ (re-exam).I really do not know know how to approach this problem properly.I have managed to prove $a^2 + b^2 + c^2 +d^2  \leq 4$ but not the initial inequality.
Show the post's comments on the "Suggested Edits" review interface Having recently reviewed a few suggested edits, I've found that not being able to see the comments on the post is hindering my understanding of the context of some edits.  In particular, there have been a few posts where someone has made a substantial edit to a question or answer which significantly alters its meaning. Typically I'd reject that as "too radical", since it can't be known that the new meaning of the post was what the original author intended.  However, in some cases it is clear from the comments that the original author is aware of the intended change, and as such the editor is not in fact misrepresenting the author's intentions. In these cases, it would have been useful to see the comments when making the review decision.  As an example, a common cause of this is when an inexperienced user is asked to clarify their question, and ends up writing a comment that would be better off as an edit to the question (due to length, excess code content, etc), presumably due to lack of awareness that the question can be edited. Often, other users will helpfully edit the question to include the information the OP put in the comment (see ,  and  for examples). In this situation, it is clear that the edit reflects the OP's intentions, but this information is only present in the comments which are not visible in the review interface.  Following is the suggested edit screen showing the suggested edit. It seems indentation is increased and one line is moved a few lines upwards and some empty lines added to increase readability. But it look like an invalid edit since changing the related line may cause alteration on the runtime. So not looks lie a valid edit. Better open the answer and examine it        Checking the Q&amp;A itself reveals that a talk had taken place between the OP and the answerer and revision done by the OP with the permission of the answerer. So it was indeed a valid edit.        In short: is there any particular reason comments are not displayed in the suggested edit review interface? I think it would add useful contextual information when making review decisions, and presumably this would be trivial to implement.  Note: this is not a dupe of , as it does not concern comments posted after the suggested edit. Find a plane that passes through a point and is perpendicular to 2 planes In reading through the original GCM specification (), the composition of the 128 bit Initialization Vector as a concatenation of a 96b nonce and a 32b unsigned wrapping counter seems arbitrary and forces the scrambling pattern to repeat every 232 16-byte block.  Is the algorithm expected to be secure for significantly longer stream lengths if for example the IV were a 128b nonce XORed with a 64b or 128b counter, or are there known cryptanalysis issues that begin to arise? Schengen area 90 days limit: Can I leave on the 91st day? Cannot inherit from multiple classes defining __slots__? What is the difference between Gödel's completeness and incompleteness theorems? I have two Ubuntu Software Centers, one can be opened, another one can't Can anyone explain reverse bias pn junction using only electrons as charge carriers In forward bias condition p-type semiconductor is connected to positive side and ntype semiconductor to negetive side and the electrons flow from n-type to p-type to cathode of battery.  But imagine the same situation when the situation is reverse biased so now p-type is connected to negetive terminal of battery and n-type to positive terminal of battery.  Now there is excess of electrons in n-region and it is connected to positive terminal of the battery.So electron can flow from n-type region to positive terminal of the battery due to potential difference created by battery whereas in p-type electrons can flow from one covalent bond to another as they do in forward biasing.  Only difference is,here electron move in opposite direction (from one covalent bond to another) i.e. from n-region to depeletion region whereas in forward bias they move from depeletion to n-region.Which shouldn't be a problem   Now in this manner current must flow regardless of biasing so what's the correction? I installed Redhat 6 x86_64. I am using the Network connection screen to set a static IP address like below (I want two PC's in my house to see each other: one Redhat PC and one Mac)   192.168.0.5   255.255.255.0   192.168.0.1     When I run ifconfig it displays only lo and virbr0 information. I don't know what these items are (I don't really know much about network settings).   When I try ifconfig -a it displays eth0, lo, sit0 and virbr0. The information for  eth0 is as follows:   Link encap : Ethernet HWaddr 90:2B:34:74:05:30 BROADCAST MULTICAST MTU:1500 Metric:1 RX packets:192 errors:0 dropped:0 overruns:0 frame:0 TX packets:6 errors:0 dropped:0 overruns:0 frame:0 collisions:0 txqueuelen:1000 RX bytes 53811 (52.5 KiB) TX bytes:468 (468.0 b) Interrupt:29 Base address:0xc000   Could someone help me to point out if anything wrong with my setting or how to resolve this problem? What is the correct way of installing BOINC software? Alarm is working even when the phone is powered off I just discovered that if I set the alarm clock and then power off the phone (by holding the power button and then selecting "Power Off") the alarm will still go off when the time comes.  Is this expected behavior? It makes me feel like "power off" doesn't actually power off the phone.  I have a Huawei Y530-U051 running android 4.3. I'm thinking that one way Jeff and the Stack Overflow team could squeeze some extra money out of this site would be to automatically convert all Amazon links posted here into affiliate links, e.g. Stick "tag=codinghorror-20" (or more likely a new site-specific tag) onto every Amazon link. This would bring in some additional revenue every time someone purchased a book via a link on this site.   They could do similar things with other links as well. Amazon's simply the most obvious choice.  So my question is, would anyone have a problem with this?  I know I wouldn't mind, but I don't know how other people would react.  What does everyone think? Is this a horrible idea, a great idea, a waste of time? I only occasionally use the Fixed Zoom In/Out buttons on the Tools toolbar of ArcMap and had always assumed that they zoomed in/out by a set scale factor.  However, using ArcGIS for Desktop 10.2, I just set my scale to be 1:10,000,000 and then hit the Fixed Zoom In button 10 times while recording the subsequent scale each time and was surprised to see:  10,000,000 8,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,500,000 2,000,000 1,500,000 1,250,000 1,000,000   Dividing each scale denominator by its successor shows a sequence of: 1.25, 1.33, 1.2, 1.25, 1.33, 1.2, 1.25, 1.33, 1.2  Curious about what happened in between the scales above I experimented a little to discover that the first issuance of the Fixed Zoom In button simply takes the scale to the first of those listed above and then starts following the same pattern.  Try starting with 1:8,000,001 to see ...  8,000,001 8,000,000 6,000,000 5,000,000 4,000,000 3,000,000   This seems like quite a neat solution, but not one that I had realised was in place - does anyone know if it has been documented anywhere?    I've looked in the Online Help without success.  In answer to @DevdataTengshe, the scales currently in my available list (see below) seem to bear no relation to those used by Fixed Zoom In/Out.    However, when I add 1:7,500,000 into my available list it does seem to insert that as an additional scale step ...  10,000,000 8,000,000 7,500,000 6,000,000 5,000,000 4,000,000   This makes the algorithm being used even more mysterious to me. Show $\sin x = bx$ has at least 3 real solutions. Show $\sin x=bx$ has at least 3 real solutions for every $b$, where $-\dfrac{2}{3 \pi} &lt; b &lt; \dfrac{2}{\pi}$.   Some useful facts I am trying to use:  Intermediate value theorem: If $f$ is a continuous real-valued function on $[a,b]$ and $d \in \mathbb{R}$ satisfies $f(a)&lt; d&lt; f(b)$, then there exists a point $c$ in $(a,b)$ such that $f(c) = d$.   Corollary: If $f$ is a continuous real-valued function on $[a,b]$, then $f([a,b])$ is a closed interval. C99 stdint.h header and MS Visual Studio How do I set a Boolean field with "yes" and "no" values? Take into account all of the reputations from all stackexchange sites In MS SQL how do I find out what tables have not been queried or updated in a long time? .NET String.Format() to add commas in thousands place for a number I want to add a comma in the thousands place for a number.     String.Format()? I'm working with some code that widely uses the idiom of returning a pointer to a static local variable.  eg:  char* const GetString() {   static char sTest[5];   strcpy(sTest, "Test");   return sTest; }   Am I right in thinking that this is safe?  PS, I know that this would be a better way of doing the same thing:  char* const GetString() {   return "Test"; }   Edit: Apologies, the function signature should of course be:  const char* GetString();