Prove by geometry or algebra $a, b$ and $c$ are the sides of right triangle where $c$ is the hypotenuse. Show that the inradius, $r$, of the right triangle satisfies $a+b+c=2r$. Circle is inscribed into the triangle.
Find the radius of the inscribed circle of a right triangle. Find the radius of the inscribed circle of a right triangle. The triangle's height is $\sqrt{6} + \sqrt{2}$ while the bisector of the right angle is 4.  This seemed like a generic similar triangles problem but it isn't that simple when I tried to solve it. Any help?  EDIT: Here is a picture of the problem
Which spells or abilities are affected by things that care about targeting? Note: this is intended to be a canonical question about whether or not a spell or ability targets so there are multiple sub-questions.   Can Hexproof or Shroud save my creature from ? What about a  that was cast for it's Overload cost? Does shroud prevent me from using the Monstrosity ability of a creature, or from giving it more counters with Proliferate? Can I cast an Aura Enchantment on one of my creatures with Shroud, or an opponent's creature with Hexproof? Or can I use the Equip ability to attach an equipment to one of my creatures with Shroud? Will 's ability trigger when my opponent plays ? And will I have to sacrifice my  if my opponent casts ? Will Shroud or Hexproof prevent an opponent from choosing me with 's ability? Will Hexproof keep one of my creatures from being affected by ? If an Aura or Equipment is attached to a creature with Hexproof, does the Hexproof protect the attached permanent? How to use the Bevel node in 2.79b use EntityFieldQuery to find user based on profile field value History substitution fails when implemented in shell script The below command works in command line  rsync -avh -r /Source/09_03_2016/ /Destination/ echo $(basename !:3)      Output 09_03_2016   But when I do this in shell script it does not work  #! /bin/bash  /usr/bin/rsync -avh -r /Source/09_03_2016 /Destination/; echo $(basename !:3)      Output  !:3 Making the bibliography appear in the table of contents As per the following example, and as in  In bash, where has the trailing newline char gone?, I want to know "why" it happens    x="$(echo -ne "a\nb\n")" ; echo -n "$x" | xxd -p  # Output is: 610a62  # The trailing newline from the 'echo' command #   has been "deleted" by Command Substitution   I assume there must be some very significant reason for a shell action, namely Command Substitution, to actually delete some data from the command output it is substituting... but I can't get my head around this one, as it seems to be the antithesis of what it is supposed to do.. ie. to pass the output of a command back into the script process... Holding back one character seems weird to me, but I suppose there is a sensible reason for it...  I'm keen to find out what that reason is... How long does it take for chat profiles to update? Following scenario:   user registers, has 1 point user visits chat, notes he cannot post because of only 1 point and needs 20 user goes back to SO, answers a question. gets enough points to chat user goes back to chat, but his profile still shows 1 point   Now my questions:   How long does it take for the profile on chat to update? Is there a way to force updating?   Update: Logging in and out did not help. $\lim_{x\to0^{+}} x \ln x$ without l'Hopital's rule I have a midterm coming up and on the past exams the hard question(s) usually involve some form of $\lim_{x\to0^{+}} x \ln x$. However, we're not allowed to use l'Hopital's rule, on this year's exam anyways.  So how can I evaluate said limit without l'Hopital's rule? I got somewhere with another approach, don't know if it's useful:   $\lim_{x\to0^{+}} x \ln x = \lim_{x\to0^{+}} x^2 \ln (x^2) = L$ $= (\lim_{x\to0^{+}} 2x)(\lim_{x\to0^{+}} x \ln x)$ $= 0 * L$   Then I just need to prove that L is finite/exists (which means it must be 0) We know that Accio spell (aka ) has some limits:   It can be blocked (e.g. Harry couldn't Accio the Diadem in RoR in Deathly Hallows). Many more failures are listed on Wikia.   But what are the exact full rules and limitations governing the spell in Harry Potter world?   Distance?  Line-of-sight?  Knowing exact or approximate location?  Specificity of item (e.g. "Accio Broom" seemed to work for Weasleys in OotP to bring their locked-up brooms - why not just a random broom?) Types of Accio-able stuff? Can you Accio water? Lump of earth? Gases/liquids/etc...? Can you Accio based on jQuery selector (or an SQL where clause), for lack of a better description? E.g. "Accio the book that has the word 'Flammel'"? "Accio biggest green ball from this room"? Can you Accio ALL items of a kind? (the reason I thought of this whole question was seeing "Accio all guns" comment in ). Can you Accio vague stuff? ("Accio meal"). Other rules that can be inferred.   NOTE: While Wikia article has some of the info, it is clearly very lacking in details on specific rules based on specific instances in canon. I'm looking for a fully comprehensive, organized ruleset for the charm, not merely 1-2 tidbits in Wikia article. What is ambient occlusion? I've heard about ambient occlusion and it looks nice, but what exactly is it? There are infinitely many prime numbers $p$ for which $p + 2$ and $p + 4$ are also prime numbers I need help with this question: There are infinitely many prime numbers $p$ for which $p + 2$ and $p + 4$ are also primes.  Where should I start? what proof techniques will be useful?  Any input will be very useful to me. I'm trying to compute the infinite series $\sum_{n=0}^{\infty} ne^{-n}$.  I know the answer is $e/(e-1)^2$, but I don't understand how to find this result.  Thanks for the help! Just curious as to how to obtain item drops from the last update.  To clarify, item drops as in weapon skins, crates, anything else Valve added without letting us know, etc.  It seems that after playing a bit, I've gotten no drops, but I've seen players who've gotten drops after a few short minutes of gameplay, but I've been playing for hours.  Is the item drop system like TF2, where you can just idle, get an item drop, come back? Is it like Dota2, where you have to be in the matchmaking system and complete matches for the chance to get drops? Or do you have to be MvP to get item drops? Is it a certain time you play the game?   Also, are weapon and crate drops independent or dependent on each other? Meaning, is the drop counter shared between Weapons and Crates, or are they both separate drop counters? Keyboard layout indicator sometimes disappears Ubuntu 18.04.1 after some update hot network questions have inconsistent CSS for visited/not visited depending on the site you're on This is a HNQ excerpt from Stack Overflow. Visited questions are objectively darker.    This is a HNQ excerpt from Super User. Visited questions are objectively lighter.    This is a HNQ excerpt from Seasoned Advice. Visited questions are objectively darker.    This is a HNQ excerpt from Puzzling, a beta site. Visited questions are objectively darker, although the difference is hard to spot.    This is a HNQ excerpt from Travel. Visited questions actually have a completely different color.    These are 5 different sites, with 5 different formats for how the HNQ sidebar displays links. Some make visited links darker. Some make them lighter. Some make them a little big darker. some give visited links a completely different color. This is very confusing: If someone visits one of these HNQ questions and wants to continue browsing the HNQ from there, they likely expect that the links in that HNQ segment have consistent layout compared to the HNQ segment from the website they came from.  Is this something worth looking into? I realize that each site has their own look and feel, but an element that appears on every site should be consistent IMHO. People have expectations about how links work, and one of those is that if you click on a link within the same domain (which the HNQ segment arguably is), they behave similary on that page as they do on the page you come from. Delete all files except specified files/folders using command line? I have a folder containing a lot of folders and different files with the following structures :  NASA │ ├── a │   ├── doc1 │   ├── doc2 │   ├── doc3 │   ├── folder1 │   └── folder2 │ ├── b │   ├── doc1 │   ├── doc2 │   ├── doc3 │   ├── folder1 │   └── folder2 │ ├── c │   ├── doc1 │   ├── doc2 │   ├── doc3 │   ├── folder1 │   └── folder2 │ ├─ x ├─ y └─ z          I want to delete the content of the folder (NASA/) except specified folders and files.   For example I want to keep a folder, b folder and x file.     I tried   :   rm !(a/) -r NASA/   And (as explained in the answer ):  find NASA/ -type f ! -iname "x" -delete   But this is not very straight forward and I have to use a bash script.    Am I missing a more easy way ? How can I do this in a single command? When did The Doctor start running so much? Thanks to Netflix Streaming, I've been watching some random classic Doctor Who stories.  One thing that strikes me as different between the modern Who and old Who was that in modern Who, they do an awful lot of running.  It's become a sort of running gag that they've referenced in-universe.  In the classic Who I've seen, he tends to drive around quite a bit instead.  In addition, The Doctor in modern Who is incredibly reliant on his sonic screwdriver, which I often don't see at all in classic Who stories.  Did the Doctor running and the sonic screwdriver get introduced over time in classic Who, or were they both added as part of the modern Who revival? Symmetry condition for continuous coordinate transformations What's the difference between Kage Bunshin no Jutsu and Tajuu Kage Bunshin no Jutsu? While the Kage Bunshin no Jutsu seems to be taught, the Tajuu Kage Bunshin no Jutsu is sealed, so nobody can learn the jutsu. Does that mean that there's a difference between these jutsus, aside from the number of Kage Bunshin created?   Or, why was the Tajuu Kage Bunshin no Jutsu sealed away? If there's no other difference, everybody could use this without reading the sealed scroll. How to "flatten" a multi-dimensional array to simple one in PHP?