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Prove this inequality on area of the triangle

Source: 2019 Jozsef Wildt International Math Competition-W. 38

May 19, 2020
inequalitiesTriangle

Problem Statement

Let aa, bb, cc be the sides of an acute triangle △ABC\triangle ABC , then for any x,y,z≥0x, y, z \geq 0, such that xy+yz+zx=1xy+yz+zx=1 holds inequality:a2x+b2y+c2z≥4Fa^2x + b^2y + c^2z \geq 4F where FF is the area of the triangle △ABC\triangle ABC
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