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exists triangle with sides \sqrt{a^2-a + 1}, \sqrt{a^2+a + 1}, \sqrt{4a^2 + 3}

Source: 2010 Saudi Arabia Pre-TST 4.2

December 28, 2021

triangle inequalitygeometryareas

Problem Statement

Let

a

be a real number. 1) Prove that there is a triangle with side lengths

\sqrt{a^2-a + 1}

\sqrt{a^2+a + 1}

, and

\sqrt{4a^2 + 3}

. 2) Prove that the area of this triangle does not depend on

a