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Question
Can you explain this math problem to me? I have the explanation, but even that I do not understand. The question is
a^2+ba+b^2=1+i
b^2+bc+c^2=-2
c^2+ca+a^2=1
(ab+bc+ca)^2=?

Does it have something to do with the complex plane? I'm confused about the explanation and have not learned this before.
http://hmmt.mit.edu/static/archive/february/solutions/2014/Algebra_Test_solution

Answer
Yes, this does have to do with the complex plane, at least in the fact that it uses complex numbers. However, its really more of a geometry and algebra problem.

Solution 1 is clever in that it recognizes that the Law of Cosines gives

x = a^2 + b^2 -2(ac)cosθ = a^2 + b^2 +ac when θ = 120 degrees, etc.

This then defines the point P inside the triangle ABC, as described. The solution then applies Heron's formula (never heard of it) to set up the rest of the equations.

Solution 2 is more how I would have tackled it, which is to say I would have used combinations of the 3 x, y, z equations to solve for ab + bc + ca as shown (I believe that "x = BC^2" should really be x = (BC)^2, etc.). The term "cyclic" just means that the companion equations for x - y, namely y - z and z - x, are defined similarly. The rest is "just" algebra.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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