Simplest known solution to "World's Hardest Easy Geometry Problem"?
"World's Hardest Easy Geometry Problem"
Which style is preferred? Is there a good reason for one vs. the other?  Thanks in advance!  1) cmds.each do |cmd|    end  2) cmds.each { |cmd|    }   Example code:  cmds = [ "create", "update", "list", "help" ]  # Block style one # cmds.each do |cmd|   puts "loop1, cmd: #{cmd}" end  # Block style two # cmds.each { |cmd|   puts "loop2, cmd: #{cmd}" } How do I connect the initial highway to my six-lane road? What issues should be considered when overriding equals and hashCode in Java? What issues / pitfalls must be considered when overriding equals and hashCode? How can I swap or replace multiple strings in code at the same time? Using awk to sum the values of a column, based on the values of another column The TweakUI powertoy for Windows XP had an option to auto-login. It stored your password and did the login for you. This was useful, since you kept password protection, but had login programs loaded when you first booted.  When you switched users or hibernated, it would still ask for a password, which is the desired behavior.  Is there a way of doing this in Windows 7? Air gap on isolation space The below photo belongs to a PSU. I wonder what was the reason of placing air gaps on the 8 mm isolation. Is it to improve the isolation or provide a thermal relief (although there is no component below the red circled air gap)? Entropy and the principle of least action Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ? how to output text to both screen and file inside a shell script? Currently I have a shell script which logs messages to a log file like this:  log_file="/some/dir/log_file.log" echo "some text" &gt;&gt; $log_file do_some_command echo "more text" &gt;&gt; $log_file do_other_command   When executing this script, there is no output to screen, and, since I'm connecting to the server via putty, I have to open another connection and do "tail -f log_file_path.log", because I can't terminate the running script and I want to see the output in real time.  Obviously, what I want is that the text messages are printed on screen and into file, but I'd like to do it in one line, not two lines, one of which has no redirection to file.  How to achieve this? Why is $2\pi i \neq 0?$ We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$  Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's formula), then shouldn't $2\pi i=0$? What am I missing here? Mortarboard badges being awarded as a result of bounties after recalc What causes this corkscrew-shaped object to rise up in a stream of water falling down? Are there consequences to stamps in my passport that are not from border control? Shortcuts in Objective-C to concatenate NSStrings Are there any shortcuts to (stringByAppendingString:) string concatenation in Objective-C, or shortcuts for working with NSString in general?  For example, I'd like to make:  NSString *myString = @"This"; NSString *test = [myString stringByAppendingString:@" is just a test"];   something more like:  string myString = "This"; string test = myString + " is just a test"; MBP 13' (mid-2010; 7,1) 2.4 Core 2 Duo P8600 supporting 16 GB RAM under Win7? I’ve got a MBP 13′ (mid-2010) 2.4 Core 2 Duo (full specs below) that I’m now operating solely under Win7 64Bit. It formerly run as well under OS X 10.6.8, relatively to which the firmware seems up to date (MBP71.0039.B0B).  Would my system recognize 16 GB RAM if operated under Win7 64 Bit? Or would it require the latest firmware (MBP71.0039.B0E) of OS-X 10.7.5 (which was never installed on my MBP) anyway?   As far as I have read elsewhere, using 16 GB RAM under OS X seems to definitely require this later version of OS X (including the respective firmware update); so I'd like to know whether this is an issue of OS X or (as I suspect) the firmware.    Modellname: MacBook Pro Modell-Identifizierung: MacBookPro7,1 Prozessortyp: Intel Core 2 Duo Prozessorgeschwindigkeit: 2,4 GHz Anzahl der Prozessoren: 1 Gesamtzahl der Kerne: 2 L2-Cache: 3 MB Speicher: 4 GB Busgeschwindigkeit: 1,07 GHz Boot-ROM-Version: MBP71.0039.B0B SMC-Version (System): 1.62f6 $a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$ $a, b,c $ are positive real numbers such that $a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$  Any ideas ? Comparing $\pi^e$ and $e^\pi$ without calculating them How can I  compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ? Proving the Cauchy-Schwarz inequality by induction Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future. I'm able to get a list of the field types with the following code.  $defs = \Drupal::service('plugin.manager.field.field_type')-&gt;getDefinitions();   I want to get the field type of a specific field. How do I get it?  For example, I'd like to check if the field type of field_image is _image. Is there code similar to the following one?  $fieldType = \Drupal::service('plugin.manager.field.field_type')-&gt;getDefinition('field_image');