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We give a simple proof of the fundamental inequality of the triangle based on calculation of the cosine of an angle in one particular triangle formed from familiar centers. For the term R (R − 2 r) which appears in the inequality we give a bound as a rational expression in terms of R and r. Then as an application we derive new refinements of Gerretsen's and Kooi's inequalities. [ABSTRACT FROM AUTHOR]

We use integral calculus to establish the inequality that π is less than 2 ϕ . [ABSTRACT FROM AUTHOR]