Given a triangle, denote by $A$ the radius of the circumscribed circle and denote by $a$ the radius of the inscribed circle. The distance between the centers of those circles is $b$. Prove that $A^2 - b^2 = 2Aa$.   I found this in a prelim paper. I really dont know where to start. (Sadly, I can only use tangent properties, arc properties and trignometry)
Consider an isosceles triangle. Let r be the radius of the circumscribed circle and p be the radius of the inscribed circle. Prove that the distance d between the centers of these two circles is $$d = \sqrt{r(2p-r)}$$
If background applications can't launch automatically how does Cardcase launch on a location change? Can I rent a camper in the US/Canada with a European B license? We are considering a holiday in North America with a camper. I have a European driver's license (B) which allows me to drive personal cars up to 3500 kg, although I can imagine that an average camper easily weighs more.  Does my license qualify me to drive a camper? If not, can I take an American/Canadian driver's test which would allow me to drive a camper? Browsing Quora, I saw the following  with contradicting answers.  For the :     The bits are represented by certain orientations of magnetic fields   which shouldn't have any effect on gravitational mass.   But,  contradicts that one:     Most importantly, higher information content correlates with a more   energetic configuration and this is true regardless of the particular   type of storage... Now, as per Einstein's most famous formula, energy   is equivalent to mass.   Which answer is correct? How to Give Credit for An Answer in the Comments I asked a question on SO and someone gave me the answer as a comment to the question. Subsequently, someone else gave the same answer as an answer.  I would like to give credit to the poster who made the comment. Is there any way to do that?   Being new to the site, my only thought would be to contact the commenter and tell him to post an answer if he/she wants credit. Imputation before or after splitting into train and test? Is it possible for an x86 processor to match an ARM processor in terms of performance per watt? From my personal experience with my tablet, and from the benchmarks and articles I've read, it always seems ARM processors, as seen in virtually all mobile devices, deliver incredible performance for the amount of power they consume.  It really seems like ARM is inherently more power-efficient than x86.  To me, it looks like that this is because x86 is a  while ARM is a . However, Intel Atom processors deliver very good performance relative to their power consumption as well.  In addition, having studied computer architecture, it is my understanding that in modern processors, the complex x86  is internally translated into simpler  that use an internal instruction set that is closer to the simpler ARM or MIPS ISAs than x86 proper.  As such, the ISA as presented to the programmer is little more than an interface to issue commands to the processor, rather than a representation of the actual low-level operations the chip performs.  Then again, implementing this translation layer requires additional silicon space on the chip...  That said, assuming that they are implemented using the same semiconductor process, is ARM inherently more efficient than x86? With a properly designed , is it possible for an x86 processor to deliver the same performance per watt as an ARM processor?  Remember, I'm looking for a well-written technical answer in the spirit of Stack Exchange and not mere speculation. What is the explicit meaning of “corresponding author”? Problem  This question was triggered by the comments on , but comments and answers on   confirm the existence of conflicting ideas.  There seem to be two prevalent interpretations of the word corresponding author:   The corresponding author is the one who corresponds the (unpublished) paper to the journal, i.e., who performs the technical process of submitting the paper, is available for requests on the review and publishing processes, and so on. The corresponding author is the one who corresponds on the (published) paper with anybody who likes to do so, e.g., because they found an error or like some clarification.   What I found out so far  Until today, I was only aware of the first interpretation, which was confirmed by my experience: For all papers I submitted, I automatically became the corresponding author and cannot remember that I could choose one of my co-authors for this role. Some of these journals do not even indicate a corresponding author to the reader. Also, there already is a way to indicate that an author is available for questions on the paper (interpretation 2): stating the e-mail address.  On the other hand, other journals do indicate corresponding authorship, which makes little sense for interpretation 1. Also, the value put into this role by some funding agencies for its  makes more sense for interpretation 2 (not that I completely agree with it).  Looking for some guidelines from publishers covering multiple fields, I found the following  from Elsevier¹, which kind-of covers both interpretations:     Clearly indicate who will handle correspondence at all stages of refereeing and publication, also post-publication.   While there are other questions on this site touching this issue, they focus on consequences of the interpretation, such as , , or .  Actual question  What is the explicit meaning of the word corresponding author? Interpretation 1 or 2 as listed above, both, something else, or is this an extreme example of  and the answer depends on the field or publisher? Please answer only if you have more to offer than personal experience and individual journal guidelines. In particular, please address the possibility of a varying meaning.     ¹ which is separate for each journal, but seems to be identical in content for many journals There is a question I have based on the fact: If you take a quadratic polynomial with integer coefficients, take the set $\{1,2,3,4,5,6,7,8\}$, make a partition $A=\{1,4,6,7\}$, $B=\{2,3,5,8\}$, and then evaluate the polynomial with the elements of each set you got an equality.  If now with a cubic polynomial and the set $\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\}$, how to pick a suitable partition that accomplish the same fact? I mean is there a general formula for bigger degree? Dependency error while installing WINE I am a Saint Lucian and will be traveling to Manchester, UK soon about which I am a bit nervous.  Run Down:   I own a small business in my country (in the Caribbean), I have a certificate mentioning the details of the business from my Government. I pay rent in my country, but I do not have any loans (I hate owing). I pay life insurance in my country. I have a US B-1/B-2 visa (Visitor &amp; Business). I have visited the US once (last year November) on my US B-1/B-2 visa.   What I will be doing in the UK:   I plan to spend three months out of the six months which would be granted if I get leave to enter stamp. I plan to skydive in the first week and spend a few nights at an Airbnb. I plan to visit the Manchester Football Club stadium, I'm not the biggest of football fans but since I'm in Manchester, why not! I plan to do some more sightseeing and enjoy my time there. After the first week I will be staying with my female friend who owns an apartment for the rest of the time showing me around and helping me experience the UK lifestyle until June.   Why I'm leaving in June:   I am a WordPress enthusiast (I'm a Programmer) and WordCamp EU will be happening in France this year so I plan to fly from the UK to Paris where I will spend three days then take another flight from there to my country.   How my flights will be:   I will be taking a one-way Thomas Cook flight from my country to Manchester but will be purchasing my ticket for Manchester to France ahead of time so I could show Thomas Cook as well as Immigration that I don't plan to stay.   My concerns, could someone address them point by point please?:   I have a bank account but I just use it for collecting easy payments from my clients and pay for expenses. I store my savings in a credit union because of higher interest rates paid than the bank. When I applied for my US visa I showed a credit union statement showing my total balance. I'm seeing online that I should provide statements but these statements would look very bad since I my bank account is just for expenses... I mean it sometimes gets as low as $30 then, when my monthly earnings come in from clients, I withdraw savings to the credit union and leave what I need in the account for expenses. I read another post and it said people with bank statements showing poor spending might have difficulty but it's simply because the account is used for expenses not savings. For the later stages of my visit (after the first week) I will be staying with my friend. I will simply be staying with her to save on cost of Airbnbs for the next two months three weeks, even though they could be cheap, free is better right? I will be booking the Airbnb, skydiving and tours prior to traveling to the UK, will that help? Is my stay too long for staying with someone? I won't be depending on her for anything since I have my own funds...it's just convenient that she has a place where I could stay. If it is then I don't have a problem shortening my stay to 1-2 weeks and just head back to my country and take a separate flight to Paris in June.   I am nervous because I really want to visit the UK and see my friend, I am also not sure if the minimum amount of time that could be given to a visitor is six months? What if I get a round trip for two weeks and the Immigration officer gives me exactly two weeks stay, is that possible? Or is the minimum they could give six months?  Your help is very much appreciated, I want soon to try backpacking (nomad life) but I don't think my poor heart could take the stress of worrying about immigration :(  Update  Since people in the comments are asking about my relationship with the friend I plan to stay with even though she's from my country originally but now a UK citizens and we are simply friends. Performance Advantages to Iterators? What (if any) performance advantages are offered by using iterators. It seems like the 'Right Way' to solve many problems, but does it create faster/more memory-conscious code? I'm thinking specifically in Python, but don't restrict answers to just that. How to find out if an item is present in a std::vector? All I want to do is to check whether an element exists in the vector or not, so I can deal with each case.  if ( item_present )    do_this(); else    do_that(); public static IQueryable&lt;TResult&gt; ApplySortFilter&lt;T, TResult&gt;(this IQueryable&lt;T&gt; query, string columnName)   where T : EntityObject {   var param = Expression.Parameter(typeof(T), "o");   var body = Expression.PropertyOrField(param,columnName);    var sortExpression = Expression.Lambda(body, param);   return query.OrderBy(sortExpression); }   Because the type for OrderBy is not inferred from sortExpression I need to specify it something like this at run time:  var sortExpression = Expression.Lambda&lt;T, TSortColumn&gt;(body, param);   Or  return query.OrderBy&lt;T, TSortColumn&gt;(sortExpression);   I don't think this is possible however as TSortColumn can only be determined during runtime.  Is there a way around this? How does an internet server respond to a request from a private IP? How to capture workload from production to replay on test? I want to perform performance testing on my test server.  How can I capture workload from production and replay on my test server?   Basically I'm looking for detailed steps so that I can replay the workload from production to my test environment. Why does subsurf modifier cause weird deformation at the end of a cylinder? My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. Pandas Merging 101 proof detail concerning bijection between a set and its power set Theorem: If $X$ is a set, then $X$ is not equivalent to its power set.  Proof: suppose for a contradiction that $f:X\to P(X)$ is a bijection. Define $B:=\{x \in X, x\not\in f(x)\}$. Because $f$ is surjective (onto), there is a $b$ in $X$ with $B=f(b)$. Now $b$ in $B$ would imply $b$ in $B=f(b)$. By definition of $B$, this would mean that $b$ is not an element of $f(b)=B$. Thus we infer $b$ is not an element of $B$. But then $b$ is not an element of $B=f(b)$, which by definition of $B$ forces $b$ in $B$, a contradiction.  I understood this proof until he reached the part that says "Now $b$ in $B$ would imply $b$ in $B=f(b)$..."  This seems problematic to me because the assumption $b$ in $B$ seems unwarranted. Since $B$ is a subset of $X$ it makes sense if we had $b$ in $B$ and then arrived at $b$ in $X$ but the reverse does not seem like a valid step and I'm left wondering if the resulting contradiction that arised came simply from the assumption $b$ is in $B$ and therefore only means that $b$ is not not an element of $B$, And that this has no bearing on the bijection part of the argument. In fact I dont even see how this connects to the part of the argument that used the onto property of bijectiveness. What are XML namespaces for? This is something that I always find a bit hard to explain to others:  Why do XML namespaces exist?  When should we use them and when should we not? What are the common pitfalls when working with namespaces in XML?   Also, how do they relate to XML schemas? Should XSD schemas always be associated with a namespace? Automating DB Restore from network share