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Let ABC be a right triangle with B the right angle. X,Y and Z are on BC, CA and AB respectively such that BXYZ is a square. If the square is of side length m, AY = r and YC = s, find m in terms of r and s.

I have two solutions for my students (I instruct a math team class): One involves proportions and the other involves the angle bisector theorem (BY is an angle bisector). I am just curious of there are any other creative solutions for this problem.

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    So this is homework!? What have you or your students already found out, since you do wanna hear things you already know...2012-07-23
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    Try Cartesian coordinates with $B=(0,0)$, $A=(a,0)$ and $C=(0,c)$. The line $AC$ is $x/a + y/c = 1$. Find intersection with $x=y$. There's more to do, but, when I had problems with contest math geometry problems, I often brute forced it this way Cartesian geometry2012-07-23
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    My students and I have done much of what's covered in math team classes: Ptolemy's Theorem, Stewarts Theorem (including a proof from Ptolemy's), Angle Bisector Theorem I and II (the latter from Stewart's), etc. I just like to show multiple solutions to my students to show them how to think creatively. I was just curious if there are any creative solutions since mine seem brute force-ish. I am currently trying to find other solutions through Stewart's Theorem, circumscribing the triangle and extending the angle bisector to use the chord theorem, amongst other things.2012-07-23

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