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In the image, is there a nice way to write down the sum of a+b+c?

enter image description here

Rory Daulton
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Desmund
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    Welcome to MathSE! You are more likely to get a good answer to your question if you follow a few guidelines. In particular, what have you tried so far, and just where are you stuck? This is not a homework-answering site: we want to see that you have put significant work into the problem. – Rory Daulton Mar 14 '16 at 23:09
  • Homework problems are on-topic here, which makes MSE effectively a "homework answering site" no matter whether the question posters add more material beyond the problem statement. Also, who is this "we"? Many users are perfectly happy with clean problem statements free of additional OP-authored content (which is usually of lower quality than the question itself, that comes from a book, class or contest authored by experts). @RoryDaulton – zyx Mar 14 '16 at 23:27
  • @zyx: I am perfectly happy to answer homework questions, if the OP shows that he is not just throwing his work to us. Many other users here agree, as shown by the quick hold placed on this question and by the upvote on my comment above. I understand exceptions when the question is quite good, and this is a classic question, but since I have seen this question often before, placing it here seems to help the math community very little. – Rory Daulton Mar 14 '16 at 23:33
  • So what is actually true is that "I, and some other frequent close voters, want to see .....". Users who disagree could actually be a majority of the main-site users. There are maybe a few dozen denizens of the meta site who are frequent closers of unadorned questions, but judging from the posting and answering activity, on MSE they are heavily outnumbered. Your comment looks like a template, and if it were edited to say "I and the regular close voters" instead or something like that, it would not speak for the silent possible-majority. @RoryDaulton – zyx Mar 14 '16 at 23:45
  • I do agree that this particular question is a duplicate, but then it would make sense to close it as a dup (which would also have the benefit of pointing the OP to the existing answers). – zyx Mar 14 '16 at 23:47
  • Related: http://math.stackexchange.com/questions/1656034 – zyx Mar 15 '16 at 00:44
  • Also related: http://math.stackexchange.com/q/1605964/409 . (In particular, my answer. :) – Blue Mar 15 '16 at 07:52
  • @Blue, that's a very nice answer. Does something like that work for the generalization with rhombi (linked there), posed at http://www.mscroggs.co.uk/puzzles/75 ? – zyx Mar 15 '16 at 20:23
  • @zyx: I'm not sure what Scroggs is expecting in his rhombic extension to the puzzle. The angle sum certainly isn't constant, and I'm not seeing a clever connection to, say, the angle made by sides of the rhombi. (The connection I've calculated is rather unattractive. Maybe I overlooked something.) – Blue Mar 16 '16 at 13:11
  • That was my impression too ---- no invariant relationship is apparent. Thanks. @Blue – zyx Mar 16 '16 at 16:33

1 Answers1

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Using the formula for the tangent of a sum: $$\tan(b+c)=\frac{\tan b + \tan c}{1-\tan b\tan c}=\frac{\frac12 + \frac 13}{1-\frac12\cdot\frac13}=1$$ so $b+c=45^\circ$. Clearly $a=45^\circ$.

grand_chat
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