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\sqrt{(a - b) (a + c)} +\sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc} isosceles

Source: 2010 Moldova JBMO TST p7

February 25, 2021
Geometric Inequalitiesisosceles

Problem Statement

In the triangle ABCABC with ∣AB∣=c,∣BC∣=a,∣CA∣=b| AB | = c, | BC | = a, | CA | = b the relations hold simultaneously a≥max{b,c,bc},(a−b)(a+c)+(a−c)(a+b)≥2a2−bca \ge max \{ b, c, \sqrt{bc}\}, \sqrt{(a - b) (a + c)} + \sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc} Prove that the triangle ABCABC is isosceles.
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