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How to prove the midpoint of a chord is also the midpoint of the line segment defined by the points of intersection of other two chords with it? Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry,  on January $29$th, $2010.$  "Let $\mathcal{C}$ be a circle, $A,B$ two distinct points on $\mathcal{C}$ and $M$ be the midpoint of the chord $[AB]$. Take two other chords,$[PQ]$ and $[SR]$, that pass through $M$. Let $C$ (resp. $D$) be the intersection of $[AB]$ with $[PS]$ (resp. $[RQ]$). Prove that $M$ is the midpoint of the chord $[CD]$."    To prove it I've written the following (failed) argument, in the  of this post (translation of mine):    The figure is symmetric with respect to $M$: $\overline{AM}=\overline{MB}$, $\overline{PM}=\overline{MU}$, $\overline{RM}=\overline{MW}$, $\overline{QM}=\overline{MV}$, $\overline{QR}=\overline{VW}$, $\overline{SM}=\overline{MT}$. From $\dfrac{\overline{SC}}{\overline{DT}}=\dfrac{\overline{CM}}{\overline{MD}}=1$ follows that $\overline{CM}=\overline{MD}$.  Here is an extract of the author's reply (translation of mine):   "It is not clear that  $\overline{QM}=\overline{MV}$. Is the point $V$ on the line $(QM)$ defined by this equality or is $V$ defined as the intersection point of the lines $(QM)$ and $(WC)$? Why are both definitions to give the same point?  Let $C^{\prime }$ be the intersection of $(SP)$ and $(VW)$, and $D^{\prime }$ the intersection of $(TU)$ and $(QR)$. (...)  One still has to show that $C^{\prime }=C$ and $D^{\prime }=D$."   I have agreed with these objections.  Until now no proof has been posted. The author considers that the "proof is not quite simple".  Q. What is the theorem this exercise refers to? Or how does one prove it? Does C1 code coverage analysis exist for Ruby? I'm currently using  to get C0 code coverage analysis for a rails project that I'm working on.  However, those results are practically meaningless- I have 100% coverage according to rcov (as it only covers C0 analysis) and I've barely written half the test cases for the functionality that exists thus far.  I'm used to the useful results from the code coverage in Visual Studio 2008 Team, which has C1 coverage.  Are there any tools that provide similar coverage for ruby? How do you configure proxy settings in the Ubuntu Server or Minimal (CLI) versions using the terminal? 2D inhomogeneous biharmonic equation How do exchanges match limit orders? Say a security has been trading at under $20.10, and there are limit orders selling at $20.10 and $20.21. Then Simon issues a limit order to buy at $20.21. According to , Simon would end up buying shares at $20.10.  Why does the exchange favor the buyer here?  I.e., why does the transaction execute at the seller's limit of $20.10 instead of the buyer's limit of $20.21?  (I.e., why don't Inara/River get to sell at $20.21 / above their limit, but Simon gets to buy at $20.10 / below his limit?)  Is the example flawed?  If it makes a difference, I'm mainly curious about NASDAQ and NYSE.  I tried Googling for specs (e.g. ), but couldn't find anything.  Only found that some  implementation executes the average of the buy/ask.  Just curious how the system works. Converting string into datetime I've got a huge list of date-times like this as strings:  Jun 1 2005  1:33PM Aug 28 1999 12:00AM   I'm going to be shoving these back into proper datetime fields in a database so I need to magic them into real datetime objects.   This is going through Django's ORM so I can't use SQL to do the conversion on insert. Importing QGIS3 processing tools in standalone script Inspired by , but for QGIS 3 (and therefore Python 3), I would like to use the QGIS processing algorithms in a standalone Python script without a GUI, with a standard installation of Ubuntu Linux. A simple import processing is insufficient: import processing processing.run(    &quot;qgis:rasterlayerstatistics&quot;,    {        'INPUT': .'/somefile.tif',        'BAND': 1,        'OUTPUT_HTML_FILE': './result.html'    } )  Results in: Traceback (most recent call last):   File &quot;&lt;stdin&gt;&quot;, line 1, in &lt;module&gt; ModuleNotFoundError: No module named 'processing'  How do you get access to the QGIS processing module to use the QGIS processing algorithms in a standalone Python script (not the QGIS interpreter)? In terminal, how can I write 'sagi' instead of 'sudo apt-get install' Can't start mongod , configured storage engine is 'mmapv1' I try to run mongo v2.6.10 on Ubuntu 16.04 lts:  mongod   And get the result:  mongod --help for help and startup options 2017-09-16T12:15:19.660+0430 [initandlisten] MongoDB starting : pid=15959 port=27017 dbpath=/data/db 64-bit host=mahdi-MS-7808 2017-09-16T12:15:19.660+0430 [initandlisten] db version v2.6.10 2017-09-16T12:15:19.660+0430 [initandlisten] git version: nogitversion 2017-09-16T12:15:19.660+0430 [initandlisten] OpenSSL version: OpenSSL 1.0.2g  1 Mar 2016 2017-09-16T12:15:19.660+0430 [initandlisten] build info: Linux lgw01-12 3.19.0-25-generic #26~14.04.1-Ubuntu SMP Fri Jul 24 21:16:20 UTC 2015 x86_64 BOOST_LIB_VERSION=1_58 2017-09-16T12:15:19.660+0430 [initandlisten] allocator: tcmalloc 2017-09-16T12:15:19.660+0430 [initandlisten] options: {} 2017-09-16T12:15:19.660+0430 [initandlisten] exception in initAndListen: 28574 Cannot start server. Detected data files in /data/db created by storage engine 'wiredTiger'. The configured storage engine is 'mmapv1'., terminating 2017-09-16T12:15:19.660+0430 [initandlisten] dbexit:  2017-09-16T12:15:19.660+0430 [initandlisten] shutdown: going to close listening sockets... 2017-09-16T12:15:19.660+0430 [initandlisten] shutdown: going to flush diaglog... 2017-09-16T12:15:19.660+0430 [initandlisten] shutdown: going to close sockets... 2017-09-16T12:15:19.660+0430 [initandlisten] shutdown: waiting for fs preallocator... 2017-09-16T12:15:19.660+0430 [initandlisten] shutdown: lock for final commit... 2017-09-16T12:15:19.660+0430 [initandlisten] shutdown: final commit... 2017-09-16T12:15:19.660+0430 [initandlisten] shutdown: closing all files... 2017-09-16T12:15:19.660+0430 [initandlisten] closeAllFiles() finished 2017-09-16T12:15:19.660+0430 [initandlisten] dbexit: really exiting now   error :     exception in initAndListen: 28574 Cannot start server. Detected data   files in /data/db created by storage engine 'wiredTiger'. The   configured storage engine is 'mmapv1'., terminating   after update mongodb show error . In the 3d-view workspace, we can use a random color to give our separated object a distinct look.  What if I don't like the result and I want to randomize the color again?   Swapping the setting doesn't change the random seed in color.    (Blender 2.80.64.) I need help to find the Galois Group of $x^5 +1$. I know that it has a 5-cycle and a 4 cycle and is not a subgroup of $A_5$. Thanks! I read the answers in this question, but they still leave my question open:   I didn't find a way in the Google Webmaster Tools to invoke a re-download of my robots.txt.  Through some error, my robots.txt was replaced with:  User-agent: * Disallow: /   And now all my content was removed from Google search results.  Obviously, I'm interested in correcting this as soon as possible. I already replaced the robots.txt, but I can't find a way to make Google update the cached version. How can I show notify-send messages triggered by crontab? When quoting something said by someone else, should I use a comma before the quote? Suppose I quote the definition given in a dictionary. Am I correctly using the the punctuation marks in the following sentences? (I am using them in the way I think correct in American English.)     One of the meanings of cat is, "a small animal with soft fur that people often keep as a pet."        A cadet is, "a young person who is training to become an officer in the police or armed forces."   Should I use a comma only when I explicitly quote somebody?     Groucho Marx once said, "Although it is generally known, I think it's about time to announce that I was born at a very early age."   Should I use a comma in this case too?     As Groucho Marx said,            No one is completely unhappy at the failure of his best friend.      On Comma Sense, A fun-damental guide to punctuation (Richard Lederer and John Shore), I read the following about using the comma:     Use commas to set off complete quotations:           The great general George S. Patton once said, "No, no—the war is this way, you idiots!" How to create a YouTube playlist from a list of links? I have very long lists of YouTube links/bookmarks. I want to create playlists for my YouTube channel with them. Is there any way to do it automatically without having to add each individual URL to a playlist? Java Object Reuse Should Java Objects be reused as often as it can be reused ? Or should we reuse it only when they are "heavyweight", ie have OS resources associated with it ?  All old articles on the internet talk about object reuse and object pooling as much as possible, but I have read recent articles that say new Object() is highly optimized now ( 10 instructions ) and Object reuse is not as big a deal as it used to be.  What is the current best practice and how are you people doing it ? Limit of $n$th root of $n$! I am asked to determine if a series converges or not:  $$\displaystyle\sum\limits_{n=1}^{\infty} \frac{(2^n)n!}{(n^n)}$$  So I'm using the $n$th root test and came up with $\lim_{n \to {\infty}}\frac{2}{n}\times(\sqrt[n]{n!})$ I know that the limit of $\frac{2}{n}$ goes to $0$ when $n$ goes to infinity but what about the $(\sqrt[n]{n!})$? During the Time War why didn't the Daleks blow up at least one TARDIS while in the Time Vortex? From the Cracks in time and space story arc it's known that if a TARDIS is blown up (possibly while in the Time Vortex) it rips creation itself apart. Also, from the Stolen Earth / Journey's End we learn the Daleks' masterplan which is subjugating creation itself via destroying it with the Reality Bomb (unfading Z-neutrino energy radiation).  In the Last Great Time War all sorts of temporal weapons were used and it's known that Daleks have the ability of time travel as well. Why didn't they use the Time Vortex back then to win the war once and for all via detonating a TARDIS while inside? What are the alternatives to OS X's Spotlight? Let $\mathbb{R^{ n×n}}$ be the space of n × n real matrices. Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$ is the sum of the determinants of $n$ matrices, in which the $i$th matrix is $A(t)$ except that the $i$th row is differentiated. Use this result to prove that the directional derivative of the determinant function at the matrix $A \in \mathbb{R ^{n×n}}$ along the direction $B \in \mathbb{R^{ n×n}}$ is given by $$(det)^{'} (A; B) = tr(Adj(A)B) = \langle Adj(A)^{T} , B \rangle $$ where $Adj(A)$ is the adjoint of $A$, and where the inner product on $\mathbb{R^{n×n}}$ is the trace inner product given by $\langle X, Y \rangle = tr(X^{T} Y )$ Conclude that $$D(det)(A) = Adj(A) ^T $$.