Prove in detail the inequality of this formula: Let x, y, z positive real number and △ABS a triangle. [ABC] denotes the triangle area and a, b, c the sides of the triangle. The inequality below is true: a2x+b2y+c2z≥4[ABC]xy+xz+yz

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Prove in detail the inequality of this formula:  Let x, y, z positive real number and △ABS a triangle. [ABC] denotes the triangle area and a, b, c the sides of the triangle. The inequality below is true:  a2x+b2y+c2z≥4[ABC]xy+xz+yz

stamptsk · Accepted Answer

Step 1ProofLet α,β,γ denote the opposite angles to the sides a,b,c respectively. R is the circumradius of △ABC. Observe that:a2x+b2y+c2z≥4[ABC]xy+xz+yza2x+b2y+c2Z≥abcRxy+xz+yzaRxbc+bRyac+cRzab≥xy+xz+yz12(4aR2x2Rbc+4bR2y2Rac+4cR2z2Rab)≥xy+xz+yzxsin⁡αsin⁡βsin⁡γ+ysin⁡βsin⁡αsin⁡γ+zsin⁡γsin⁡αsin⁡β≥2xy+xz+yzxsin⁡(π−α)sin⁡βsin⁡γ+ysin⁡(π−β)sin⁡αsin⁡γ+zsin⁡(π−γ)sin⁡αsin⁡β≥2xy+xz+yzxsin⁡(α+β+γ−α)sin⁡βsin⁡γ+ysin⁡(α+β+γ−β)sin⁡αsin⁡γ+zα+β+γ−γ}{sin⁡αsin⁡β}≥2xy+xz+yz

Eliza Norris · Accepted Answer

Step 1  Here is my algebraic proof.   We need to prove that:  (a2x+b2y+c2z)2≥∑cyc(2a2b2−a4)(xy+xz+yz)  or  c4z2−((∑cyc(2a2b2−a4)−2a2c2)x+(∑cyc(2a2b2a4)−2b2c2)y)z+  +a4x2+b4y2−(∑cyc(2a2b2−a4)−2a2b2)xy≥0  for which its

RizerMix · Accepted Answer

Step 1 Writing the expression as a2x+b2y+c2zxy+yz+zx≥4[ABC] the RHS is independent of x,y,z, therefore it is necessary and sufficient to prove the inequality in the worst possible case, i.e. when the LHS is minimized in x,y,z. In particular, by homogeneity we can fix a,b,c and consider the problem min{a2x+b2y+c2z:xy+yz+zx=1, x, y, z≥0} If two of x,y,z are zero the inequality is trivially proven. If just one of them is zero, say z, then the problem becomes min{a2x+b2y:xy=1, x, y≥0}=2ab by AM-GM, and clearly 2ab≥2absin⁡γ=4[ABC] The last case is x,y,z&amp;gt;0. By the Lagrange method we obtain a critical point in the interior where (a2, b2, c2)=λ(y+z, z+x, x+y) that is x=b2+c2−a2 y=c2+a2−b2 z=a2+b2−c2 up to a multiplicative constant. Substituting above we obtain a2x+b2y+c2z=2(a2b2+b2c2+c2a2)−(a4+b4+c4)=16[ABC]2 by Herons

Prove in detail the inequality of this formula: Let x, y,

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