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One of my students found this problem and gave it to me in order to help him but I cannot really think of something that will help for the solution. So, this is the problem:

ABC is a random triangle. We build 3 triangles ABD, BCE and ACF with the given angles. Find the angles of triangle DEF.

enter image description here

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closed as off-topic by Andrew Sanfratello, Brendan W. Sullivan, Bill Dubuque, vonbrand, Tommi Brander May 23 '18 at 20:08

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    $\begingroup$ This belongs on math stackexchange instead. $\endgroup$ – Brendan W. Sullivan May 23 '18 at 15:28
  • $\begingroup$ I know but I asked too many questons there so ... yeah I agree. $\endgroup$ – Maxim Kasnedelchev May 23 '18 at 15:35
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I just did a quick geogebra mockup to confirm my suspicion. There is a two parameter family of possible angles for $\triangle DEF$. Below are two examples. You can think of the two parameters as either the two parameters of the angles of $\triangle ABC$ or as the coordinates of $C$.

You should be able to write down the angle relations for each triangle and angle sums at points to get a system of linear equations. Once solved, you should get an explicit 2-parameter family of solutions dependent on any pair of the angles from $\triangle ABC$.

A triangle

Another triangle

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