Suppose that M,N, and P are the midpoints of sides AB, AC, and BC of triangle ABC. Show that the centroid of triangle MNP is the same point as the centroid of triangle ABC.
The question is as follows:      Let $A'$, $B'$ and $C'$ be the midpoint triangle of triangle $ABC$. In other words, $A'$, $B'$ and $C'$ are the midpoints of segments $BC$, $CA$ and $AB$, respectively. Show that triangles $A'B'C'$ and $ABC$ have the same centroid.    My reasoning was through the means of the Midline Theorem. After drawing the diagram, we can see, for example, $C'A'$ is a midline of $\triangle ABC$, which means that $C'A'$ is parallel to the base $AC$ and half the length of it as well. Therefore, the median $B'B$ would pass through the midpoint of $C'A'$, including the midpoint of $AC$.   Is my reasoning valid or, at least, is it on the right track? Any advice on solving this problem will be greatly appreciated.
Stack exchange badges It might be useful to encourage existing users of stackoverflow to cross pollinate other stack exchange sites.  While rep-sharing is not useful, as rep is meant to be a measure of expertise in one subject, badges may be one way to encourage participation across many sites.  This was triggered by seeing how few people actually had greater than 1k rep on all three main sites (SO, SF, SU) via Dennis Williamson's  app.  I had tried to do this by hand prior to the API under the assumption that many people would be answering questions on all three sites, but it turns out there are only 33 people, of all the users of stackoverflow, that have a 1k+ reputation on all three sites.  Over time it may be useful to provide badges that appear on all sites one is participating in that are awarded for extraordinary participation in the whole universe of Stack Exchange sites.  For instance, a silver badge might appear on all my profiles if I obtain 1k rep on 5 or more sites, and a gold badge if I have more than 3k rep on 5 or more sites.  I would hesitate extending this much past 5 sites as we don't want to encourage spamming questions and answers on sites where the user has little interest, but across the universe of stackapps I know I'm proficient or expert in at least 10 current sites, and while it's not a big nudge, the idea that I can get extra awards - especially ones that appear on all sites I'm in - would move me to participate a little more on sites that I am registered on, but don't visit regularly.    I expect, eventually as sites are launched, most users of any one site will have some interest or knowledge in at least 5 other sites.  This may provide the extra nudge some sites need to maintain an excellent rating in terms of usage. Fubini's theorem for series with dependent indices As a part of a proposition about power series that I am trying to prove I have to show that  $$ \sum_{m=0}^{\infty} \left(\sum_{n=m}^{\infty} \frac{n!}{m!(n-m)!}(b-a)^{n-m}c_n\right)(x-b)^m = \sum_{n=0}^{\infty}c_n(x-a)^n$$  Proof: What I have done so far is to show that  $$\sum_{m=0}^{n} \frac{n!}{m!(n-m)!}(b-a)^{n-m}(x-b)^m = (x-a)^n $$  So the next idea is to use Fubini's rearrangement theorem for infinite series (I have also shown that the series are absolutely convergent). However, the theorem does not accomodate for dependent indices. Intuitively (using triangular summation), I think that the proper way to interchange series would be \begin{equation*} %\begin{array}{ll} \sum_{m=0}^{\infty} \sum_{n=m}^{\infty} \frac{n!}{m!(n-m)!}(b-a)^{n-m}c_n(x-b)^m = \sum_{n=0}^{\infty} \sum_{m=0}^{n}\frac{n!}{m!(n-m)!}(b-a)^{n-m}c_n(x-b)^m %\end{array} \end{equation*} and thus  $$=\sum_{n=0}^{\infty} c_n(x-a)^n$$ However, I do not know how to perform this step rigorously in case of infinite summation. Which tools and technologies are used to build the Stack Exchange Network? What tools and technologies are used to build the Stack Exchange Network?  See also: How to access a usb flash drive from the terminal? I only have access to the command line. I need to backup my data (on the user folder) to a pen (USB DOK).    How can I mount a flash drive manually? What should be the copy command? WOL was working great on Bionic until yesterday's kernel update. How to diagnose/recover it?  Edit: if set to boot from 4.15.0-23 kernel it works. If set to boot from 4.15.0-24 it does not.  Edit2: WoL can be enabled with   sudo ethtool -s [card] wol g   but it isn't persistent. Searching for how to set it every boot.  Edit3: Per a comment by Kai-Heng Feng on this  this was an advertent change! (and therefore not a bug)  This is because the fix of LP: #1752772 has one commit [1] that disables WoL by default. Set WoL explicitly should solve your issue.  [1] https://github.com/torvalds/linux/commit/7edf6d314cd061e1d0a1b7bc0b511d64322c3f72 Table cells larger than they are meant to be I've created a map system for a game that runs on the principle of drawing the picture of the map from tiles. There are many reasons for this which I won't go into here but if you really want to know then I'm sure you can find out how to contact me ;)  I have  so you can see exactly where the problem lies and the source. The issue is the line between the top 2 tiles and the bottom 2 tiles, I can't figure out why it's gone like this and any help would be appreciated.  In the source is a marker called "stackoverflow", if you search for "stackoverflow" when viewing source then it should take you to the table in question.  I have also uploaded an . How do I disable notifications in my personal profile for activity on pages I am an admin of? I'm an admin of several pages on Facebook (for technical stuff like RSS imports, etc.), but neither do I contribute content nor do I care about their content.  Still, as an admin, I get notifications (upper left hand, the globe icon) about activity (someone liked this, someone posted that) on those pages in my personal (!) profile .  Does anyone have the same issue? Is there any way to explicitly disable notifications for certain pages?  (This behavior seems to be new, by the way; I'm only seeing those notifications for a few weeks now.) I had booked a ticket for travel through Indian Railways using IRCTC and got a confirmed ticket. While booking, I forgot to mention the details of a one year old child who will also be traveling.     Will it create any problem while boarding the train? Persist sort/filter selections Everybody perhaps knows of Jon Favreau's movie  which is going to be released in 2016. But quite recently I learned (and to my utter surprise) that another Jungle book movie is coming in 2017 and it's called .   Now, this movie has big names too with Christian Bale as Baghera, Benedict Cumberbatch as Sher Khan, Cate Blanchett as Kaa and a young Indian boy in the live action role as Mowgli.   How often does this happen, with two big productions with big cast making a movie on the same subject? I want to ask why this is happening and if there are any really significant differences between the two (apart from the cast, of course). What does "causally connected" or "causes" really mean? In a different thread, a user stated the following about events preceding or following other events:     However, if the two events are causally connected ("event A causes event B"), the causal order is preserved (i.e. "event A precedes event B") in all frames of reference.   My question is what does "causally connected" really mean? What does "causes" mean? Further, given that we know that we can have instantaneous effects in typical quantum processes (e.g. flip the polarizer, effect on another a reading light years away, even though we cannot transmit useful information with this), does that not constitute "causing" for the purpose of this statement? In the following example, is it appropriate to use a or an as the indefinite article, and why?     He ate __ green apple.   I know that in the case of just "apple", it would be "an apple," but I've heard conflicting answers for "green apple," where the noun is separated from the article by an adjective.  Also, which is more appropriate in this case:     He ate  __ enormous Pop-Tart. How to set a custom 4x4 projection matrix in Blender Why do corticosteroids harm COVID-19 patients? states that steroids could do more harm than good in treating coronavirus (COVID-19), referring to .      Understanding the evidence for harm or benefit from corticosteroids in   2019-nCoV is of immediate clinical importance.   It's unclear to me how exactly the use of corticosteroids (which is a common treatments for Asthma patients) can do harm other than the vague explanation of "steroids also impair the immune system's ability to fight viruses".   How exactly does the use of steroids harm the body in case of fighting a virus such as SARS-CoV-2 (2019-nCoV) that causes COVID-19? Why intra-word ** highlighting is broken to protect programming_underscore_usage? htl5 voice recording with isualization I have searched this for hours but still no luck, obviously doing something wrong.  The journal (Risk Analysis) wants "Reference style is based on the National Library of Medicine (NLM) style guide. In-text citations are numbered, superscripted and in parentheses, and the reference list is numbered in order of appearance within the manuscript. Footnotes are superscripted without parentheses".  Could someone please suggest an appropriate .bst file to use that will produce this? Thanks, J How do I enter a strong (long) DKIM key into DNS? Should a toggle button show its current state or the state to which it will change? How to avoid tripping over UTF-8 BOM when reading files I'm consuming a data feed that has recently added a Unicode BOM header (U+FEFF), and my rake task is now messed up by it.  I can skip the first 3 bytes with file.gets[3..-1] but is there a more elegant way to read files in Ruby which can handle this correctly, whether a BOM is present or not?