If (x+y):(y+z):(z+x)=3:5:6 and x+y+z=14, then what is x^{2}+y^{2}+z^{2} equal to ?

Explanation

Let x+y=3k, y+z=5k, and z+x=6k. Adding all three gives 2(x+y+z)=14k. Since x+y+z=14, 14k=28 \implies k=2. Thus x+y=6, y+z=10, z+x=12. Solving yields z=8, x=4, y=2. x^2+y^2+z^2 = 16 + 4 + 64 = 84.

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