MathDB

Triangle inequality

Source: Romania TST 2015 Day 1 Problem 2

April 9, 2015

circlestriangle inequalityinradiusinequalitiesgeometry

Problem Statement

Let

ABC

be a triangle, and let

r

denote its inradius. Let

R_A

denote the radius of the circle internally tangent at

A

to the circle

ABC

and tangent to the line

BC

; the radii

R_B

and

R_C

are defined similarly. Show that

\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}