Let and be the lengths of the sides of a nondegenerate triangle, let , and let and be the inradius and circumradius of the triangle, respectively. Show that
and determine the cases of equality.
A)Proof of the 1st inequality:
Assume that is area of the triangle, then and , therefore that inequality is equivalent to
Setting , then and we need only prove
By AM-GM we have and therefore and also 1st inequality proved.
Equality occur iff iff triangle is equilateral.
B)Proof of the 2nd inequality:
By AM-GM we have
and we’re done. Equality occur iff iff .
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