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Geometric inequality for radii of incircle and circumcircle

Source: 2003 Junior Macedonian Mathematical Olympiad P3

June 29, 2021

geometrygeometric inequalityinequalities

Problem Statement

Let

ABC

be a given triangle. The circumcircle of the triangle has radius

R

, the incircle has radius

r

, the longest side of the triangle is

a

, while the shortest altitude is

h

. Show that:

\frac{R}{r} > \frac{a}{h}