Prove that $r_1^2+r_2^2+r_3^2+r^2=16R^2-a^2-b^2-c^2$  Where  r denotes the respective in-radius and R the circum radius  a,b,c are the sides of the triangle    I have tried using the   $\frac{\Delta}{(s-a)\dots}$  But I get a fraction with a irreducible $\frac{denominator}{numerator}$.
Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof would be super cool. I did prove it but it took me ages to do the calculation. Thanks.
How do I reverse a list using recursion in Python? I want to have a function that will return the reverse of a list that it is given -- using recursion. How can I do that? Wrong numeration in toc longer then one page Is it safe to set xp_cmdshell value 1? I need to use xp_cmdshell in one of SQL job.So is it safe to reconfigure xp_cmdshell to 1 or can i set value 1 in the starting of SQL statements and set it back to 0 in the end of SQL Statement ? I have a set of polygons representing administrative districts.  The districts were digitized by hand, and there are small spaces (slivers) between polygons where the polygons should be touching. How can I have the polygons snap together, removing the spaces? How to rotate the curve but not the axes? Cycles not rendering image or movie clip used as background in the viewport How to get all possible combinations of a list’s elements? I have a list with 15 numbers in, and I need to write some code that produces all 32,768 combinations of those numbers.   I've found  (by Googling) that apparently does what I'm looking for, but I found the code fairly opaque and am wary of using it. Plus I have a feeling there must be a more elegant solution.  The only thing that occurs to me would be to just loop through the decimal integers 1–32768 and convert those to binary, and use the binary representation as a filter to pick out the appropriate numbers.   Does anyone know of a better way? Using map(), maybe? I have to format my entire disk and reinstall Ubuntu.  I have installed a lot of software on my current system. And I will have to reinstall all those updates, drivers and applications too.  Is there any way I can make an .iso image of my installed Ubuntu along with all the software, preferences, drivers etc in it ?  Is there any way I can back up my whole OS along with all its software, drives, updates and all ? Is $(x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}}$ Differentiable in $(0,0,0)$? $$f(x,y,z)=\begin{cases} (x^2+y^2+z^2) \ \sin \frac{1}{\sqrt{x^2+y^2+z^2}} \qquad (x,y,z) \ne (0,0,0) \\ \\ 0 \qquad (x,y,z)=(0,0,0) \end{cases} $$   At first, I study the continuity in the origin.  I apply the concept of sequential continuity: $x_n \rightarrow x_0 \Rightarrow f(x_n) \rightarrow f(x_0)$   So, $x_n=\frac{1}{n}  $ $$x_n \rightarrow 0 \Rightarrow f(x_n,x_n,x_n) \rightarrow f(0,0,0)=0$$ $$\lim_{n\rightarrow +\infty} \frac{3}{n^2} \ \sin \frac{1}{\frac{\sqrt{3}}{n}}=0 $$  How can I continue the study of differentiation?  Thanks! How to show this function's discontinuity?  $ f(n) = \left\{     \begin{array}{l l}      \frac{xy}{x^2+y^2} &amp; \quad , \quad(x,y)\neq(0,0)\\      0 &amp; \quad , \quad(x,y)=(0,0)    \end{array} \right.$ I used osm2pgsql to import data into my PostgreSQL database and it put gave me the following 4 tables, planet_osm_point, planet_osm_line, planet_osm_polygon and planet_osm_roads.   I have looked at the schema for osm2pgsql and inside the planet_osm_point table there is a column 'way' of type point. I tried to query it as an array like it says in the documentation here ():     It is possible to access the two component numbers of a point as   though the point were an array with indexes 0 and 1. For example, if   t.p is a point column then SELECT p[0] FROM t retrieves the X   coordinate and UPDATE t SET p[1] = ... changes the Y coordinate. In   the same way, a value of type box or lseg can be treated as an array   of two point values.   Anyone know the correct query the planet_osm tables from osm2pgsql based on long/lat? For example if I wanted to retrieve the information for Big Ben (London) I would query along the lines of:      SELECT * FROM planet_osm_point      WHERE ST_Y(ST_Transform(way, 4326)) = '-0.12'        AND ST_X(ST_Transform(way, 4326)) = '51.5';   There is no error message, it comes out with the table names, but no data is output either. (Its an empty table). What am I doing wrong here?  Perhaps it is too precise, i.e. there are no coordinates that match it exactly? Perhaps I need to try being more general, say 51 &lt; long &lt; 52 and -0.5 &lt; lat &lt; 0. Although this would throw back multiple results so would not be quite as accurate. I like to edit my answers. To clarify, illuminate, add examples, add links, point to new information as it crops up, etc.  It's highly irritating that my effort to improve my answer turns it into a Community Wiki.  Could anything be done to keep me from turning into a collective? Output for Print Is there a way to output for Print. I want to save in a higher DPI than the default 72dpi say 300dpi that's a good quality for printed pieces. I tried downloading an add-on but don't know if I am installing it correctly it isn't showing up but the bar in python is green. I'm using the 2.69 version of Blender. If no such feature exists what size have any of you found best for print quality. Thank you I am using Ubuntu 16.04, with Gnome 3.18. Using extensions I have gotten rid of the dash panel and only use the Applications Menu in the upper left-hand corner of the screen. (I don't have Unity.) What I can't figure out is how do you add a new program, move software around to different categories, and create new categories?  Also, any keyboard shortcut to access the Applications Menu? What is the importance of eigenvalues/eigenvectors? What is the importance of eigenvalues/eigenvectors? I need some advice as I am struggling with the following combinatorics exercise.  Let $k$ be a given positive integer. Show that any non-negative integer $N$ can be written uniquely in the form $$N=\binom{x_k}{k}+\binom{x_{k-1}}{k-1}+\dots+\binom{x_1}{1},$$ where $0 \leq x_1 &lt; \dots &lt; x_{k-1} &lt; x_k$.  The exercise is accompanied by this advice:  Let $x$ be such that $\binom{x}{k} \leq N &lt; \binom{x+1}{k}$. Then any possible representation has $x_k = x$. Now use induction and the fact that $N-\binom{x}{k} &lt; \binom{x}{k-1}$ to show the existence and uniqueness of the representation.  I am really stuck and thus grateful for the slightest hint on how to solve it. How to find all subgroups of a direct product? I am wondering how do we find all subgroups of a direct product? Is there a method to find it?  For example, how can we find all the subgroups of $\mathbb{Z}_2\times\mathbb{Z}_2$?  There is the answer: $\{(0, 0), (1, 0)\}, \{(0, 0), (0, 1)\}, \{(0, 0), (1, 1)\}$, and the improper and trivial subgroup. But I do not know the method how to find it.   All I can think about is the following: since the order of $\mathbb{Z}_2\times\mathbb{Z}_2$ is $4$, the order of a subgroup of it can be $1,2$ or $4$.   If order is $1$, then it is trivial, and if order is $4$, it is proper. Then, the order of a subgroup is $2$. We know all the elements of   $\mathbb{Z}_2\times\mathbb{Z}_2$, so we try all the pairs.   Is that the method?   But what if we are given a bigger order product, such as $\mathbb{Z}_2\times\mathbb{Z}_8\times\mathbb{Z}_3\times\mathbb{Z}_9$? It looks very long to try all possibilities.   Can anyone help?  Thanks Reading Excel files from C# SemanticImport vs Import differ on same CSV file? How can I use Netflix on Ubuntu? Is there a way to stream Netflix content to my Ubuntu machine?