The beauty of math

You are given the triangle on the in the figure below. A friend of mine claims that she can inscribe a square in the triangle -- that is, that she can find a construction, using straightedge and compass, that results in a square, all four of whose corners lie on the sides of the triangle. Is there such a construction -- or might it be impossible? Do you know for certain there's an inscribed square? Do you know for certain there's a construction that will produce it?





Note that there are several steps to this problem. The first is to prove such a thing exists (hint: It does!). It is also interesting to prove if this holds for all triangles. The next step, finding if/how you could construct this square using only a compass and straightedge, is more complicated, but it is an example of the true beauty of math. It leads to enlightenment and turns out to be a better high than most drugs. Solutions will be posted in the next post, but I'll put them in the reply section so as not to ruin anything for anyone. The solution is beautiful even without struggling with the problem and trying to solve it yourself, but you will appreciate its beauty more if you engage with the problem a little ahead of time. Also, you may find an alternate solution to the one I will post (there are many), which is also satisfying, (though I think you would be hard pressed to find something more elegant that this one which someone else came up with in class).