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I am currently starting to teach math to highschool kids who don't do well on their own, and I'm teaching a guy about notable points and lines of a triangle (incenter, centroid, circumcenter, perpendicular bisector, etc) and since we already finished with all the definitions and properties that he needs for his test, I want him to be able to solve problems related to it in case he faces them in the test.

I gave him an excercise about a basketball court with three backboards, in which he had to choose the optimal place to get as many points as possbile in each one without moving, and one about a triangular room in which a pool was going to be placed, and he had to figure out the maximum size possible for it.

Both of these examples seem a bit unlikely to happen in reality and after searching a bit I was unable to find practical uses for these tools.

Any sample excercise you can come up with will be much appreciated.

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I love this example. Suppose you want to cut out an irregular triangle from the center of a piece of paper. You can do it with one straight cut, first folding flat along angle bisectors:

Fig5.7          
And of course the angle bisectors meet at the center of the incircle (Proposition 4, Book IV of Euclid):
          Fig5.8
I have found it quite effective to hand out sheets of paper with a large irregular triangle predrawn, and have students fold & cut with scissors. It makes the incenter tangible.
These figures are from How To Fold It: The Mathematics of Linkages, Origami, and Polyhedra.

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  • $\begingroup$ I don't have this book, but what is the claim being made in part (a) of figure 5.8 above? $\endgroup$
    – Nick C
    Nov 9, 2017 at 1:50
  • $\begingroup$ @NickC: That it is at least conceptually possible that the angle bisectors do not meet in a point. But when you fold them carefully they do meet in a point. $\endgroup$
    – Joseph O'Rourke
    Nov 9, 2017 at 1:51
  • $\begingroup$ I should do this myself first to actually be sure to understand what's going on, but from what I think I got this just proves that the incenter exists, not that it can be useful in a real-world-problem. $\endgroup$
    – Otomeram
    Nov 9, 2017 at 19:17
  • $\begingroup$ @Otomeram: If the bisectors did not meet at the incenter, then those folds would not suffice for one-cut. The folds would be more complicated. As they are for an irregular quadrilateral. What constitutes a real-world problem is of course in the eye of the beholder. :-) $\endgroup$
    – Joseph O'Rourke
    Nov 9, 2017 at 19:48
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    $\begingroup$ This is a great example! And (I think Joseph O'Rourke knows this), it's isomorphic to an actual real world problem, it turns out -- if you had a shed whose floor is triangular and wanted a non-flat roof (but with each roof segment having the same pitch), the ridges would occur along these fold lines. This is also true for arbitrary polygons that you wanted to fold onto a single line. At least, that's my understanding. See the Wikipedia entry on straight skeleton. $\endgroup$
    – pjs36
    Nov 9, 2017 at 23:56

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