Triangle ABC is isosceles with BC as base, AB=AC and Angle A=20 degrees. Points D and E lie on sides AB and AC respectively, such that D lies between A and B, E lies between A and C, angle BCD=50 degrees, angle CBE=60 degrees. Determine Angle ADE.  Edit: I have tried to solve this question with simple additions/subtractions of angles with and without additional geometric constructions to achieve the goal but in vain. Is there a solution that does not involve trigonometry?
The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it.  So here is the puzzle:    The triangle $ABC$ is known to be isosceles, that is, $AC=BC$. The labelled angles are known to be $\alpha=\gamma=20°$, $\beta=30°$. The task is to find the angle labelled "?".  The only solution that I know of is to use the sine formula and cosine formula several times. From this one can obtain a numerical solution. Moreover this number can be algebraically shown to be correct (all sines and cosines are contained in the real subfield of the 36th cyclotomic field). So in this sense I solved the problem, but the solution is kind of a brute force attack (for example, some of the polynomials that show up in the computation have coefficients > 1000000). Since the puzzle originates from a book that deals only with elemetary geometry (and not even trigonometry if I remember correctly) there has to be a more elegant solution.
Can't install MySQL on Mountain Lion What is the next perfect square of the form 14444... in decimal notation? We know that $12^2 = 144$ and that $38^2 = 1444$. Are there any other perfect squares in the form of $\frac{13}{9} (10^n - 1) + 1$ (i.e. $1$ followed by $n$ $4$'s), and how would we prove it? I'm subscribed to the Ubuntu Security Notes, so everytime that I receive an e-mail, a new update is avalable.  But in the last update, they wrote something in the end that I coudn't undertand much, so can anyone please tell me what does that means?  ATTENTION: Due to an unavoidable ABI change the kernel updates have been given a new version number, which requires you to recompile and reinstall all third party kernel modules you might have installed. If you use linux-restricted-modules, you have to update that package as well to get modules which work with the new kernel version. Unless you manually uninstalled the standard kernel metapackages (e.g. linux-generic, linux-server, linux-powerpc), a standard system upgrade will automatically perform this as well.   I tried to ask to my friend (who also uses Linux), but he don't know much about kernel. How can editors and reviewers detect data manipulation? Has the Magical Community breached the Final Frontier? (aka Wizards in Space?) Have they? Given that Muggles put the first man in space in 1961, even by 1991 a lot of Muggleborns perfectly aware of the space-travel would have passed through Hogwarts and into the Ministry, even the Department of Mysteries (which seems to be a bit like their research division/Area 51 combo) And even leaving Muggle technology (which Wizards don't care for or about) aside, Humans have wondered about air travel to &quot;other worlds&quot; and meeting aliens from the time of the Ancient Greeks, surely some Witch or Wizard somewhere might have wondered if it really was possible to fly to the moon? So, is there any canon (aka Books + Pottermore + Rowling interviews) evidence of the Wizarding community having ever travelled or trying to travel to space / other planets or satellites? ETA: Note, I am also totally okay (and in fact would be eager for) historical &quot;Famous Wizard&quot; card-like tidbits from canon-sources.  AWESOME example (that I'm frankly not expecting): When the Muggles managed to land on the Moon, the Department of Mysteries had to scramble to hide the Silver-backed Moon Dragon conservatory they had been maintaining near the landing site. ALSO AWESOME example (that I am hoping for): Warlock Flibberti and Witch Tigget misused this earth-based spell and amusingly found themselves on the moon with their pet monkey instead! The title pretty much says it all: how can I type Unicode characters - specifically the em-dash —? Someone gave me a bunch of envelopes of pizza crust yeast, "specially formulated for pizza crust". The packets also say that no rise time or proofing is needed, and that it's "NOT recommended for bread baking." The ingredients are yeast, enzymes, sorbitan monostearate, l-cysteine, and ascorbic acid. It does provide a recipe for pizza crust, with just mixing and a bit of kneading.   What has actually been done to this yeast to make it work like this? (What are those enzymes?) And is there any way I can use it for other things, or is it really only good for things like pizza crust? I am not sure what caused this error, but here is what the whole error says, and also this is sitting as a notification, and preventing me from updating any software using Update Manager - Please provide some assistance or tell me how to figure out what to do to fix it.  Could not calculate the upgrade  An unresolvable problem occurred while calculating the upgrade.  Please report this bug against the 'update-manager' package and include the following error message: 'E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.' Which is more correct:     Apropos of your earlier comment, I decided to....   or      Apropos your earlier comment, I decided to...   Actually, apropos is so fancy a word, even I, a word maven if I do say so myself, am not entirely sure when to use it.  If I may add to my preposition question, are there any constructions where it would be appropriate to use something like with regard to and not apropos? Extended sound of thunder Turn off text message (SMS) sync to Exchange email account I recently started using a Samsung Galaxy S4 and one of the features in a synchronization between text messages and email accounts.  When a text message comes in it throws a copy into my Exchange email accounts that are one the phone. I want to turn this feature off but cannot find a setting for it anywhere.  Any ideas? Is there a way to transfer more than one Pokemon at a time? Is there a way to transfer more than one Pokemon at a time to the professor?  Or possibly an addon that gives me this ability?  Vendoring all the trash Pokemon for candies individually is really annoying and time consuming. If a Matrix Has Only Zero as an Eigen-Value Then It Is Nilpotent How to make git ignore changes in case? I'm not too sure what is going on here, but sometimes a particular file in my repository will change the case of its name. e.g.,: before: File.h after: file.h I don't really care why this is happening, but this causes git to think it is a new file, and then I have to go and change the file name back. Can you just make git ignore case changes? [edit] I suspect it is Visual Studio doing something weird with that particular file, because it seems to happen most often when I open and save it after changes. I don't have any way to fix bugs in VS however, but git should be a bit more capable I hope. sum pair of columns based on matching fields I'm on 10.9, and I had been using internet sharing successfully again for the first time since Lion. Today though, it just decided to stop working again.  Network layout:  Modem -ethernet-&gt; Router -ethernet-&gt; Mac Pro -ethernet-&gt; MacBook Pro Retina   The internet works fine on the Mac Pro, but it is no longer forwarding the proper DHCP settings to the MacBook.  Previously, the MacBook would get the IP address of 192.168.2.12, and was able to surf the web fine. (Though it wasn't able to connect to any AirPlay/AirPrint devices while on the ethernet interface)  Sometime today, a setting got messed up, and I'm not sure how to fix it or what setting it is.  The MacBook now has a "Self-assigned" IP of 169.254.121.24, even though it is set to use DHCP. The ethernet port on the Mac Pro that the MacBook Pro is connected to is showing a "Self-assigned" IP of 169.254.36.53, even though it is set up to use DHCP as well.  The internet sharing options appear to be correct: share connection from Ethernet 1 to Ethernet 2.  But alas, for no apparent reason, I'm not getting internet access. "No, I didn't see him" vs "Yes, I didn't see him" Bob didn't come to office today.  I was asked "Didn't you see Bob today?"  Should I say "Yes, I didn't see him today" or "No, I didn't see him today"? Including a relative path to a class file I want to model the transfer function of a resistor mathematically, and am after some resources / tips on if it's possible to do so.  For example, if I put 1V into a 1 Ohm resistor, I want to know how long and how (shape) it will take to reach a steady state temperature, given 25 deg C ambient.  I.e.   $$ \frac{T(s)}{V(s)} = ??? $$  Where T(s) is the laplace transform of temperature T(t), measured in say Kelvin.  I understand there are a lot of factors which will affect this model (material, size, shape, ambient (?)), but I'm not sure where to even start. A  shows the heat rise at a certain load, but not really how it gets there. I can't find a time-domain function for the temperature rise either. This is an exam question I was practising. I have the general understanding of Linear programming, but how would you go about finding the Decision Variables, Objective function and Constraints for this?  My Opinions: Decision variables: $M_1$- # of Model $1$ to be produced $M_2$ and $M_3$(similar definition as $M_1$)  Objective function: I'd Multiple the Cost to make in house by cost to buy from subtractor  to get Minimise: $3900M_1+7500M_2+27300M_3$  Constraints: $3M_1+1.5M_2+2M_3\ge0$  $1.25M_1+2.5M_2+0.75M_3\ge8500$  $M_1,M_2,M_3\ge 5500$  No idea where number order would go? Do I use it in the Constraints and multiply by the Hours?  Now, am I on the right track?