Question

In an equilateral triangle the cirum radius is n times inradius then n is equal to

Answer 1

For a given perimeter, an equilateral triangle uniquely maximizes the inradius and minimizes the circumradius. Hence it is the only triangle in which R=2r, and all other triangles have R>2r.

Let a,b,c are the sides of a triangle, A= area of the triangle, s= semi-perimeter.  

R=abc/4A,r=A/s

We have to show R≥2r.  

The relation abc/4A≥2A/s holds  

if abc≥8A/2s

if abc≥8(s−a)(s−b)(s−c)

if abc≥(b+c−a)(c+a−b)(a+b−c)

This is true for all triangles.  

When a=b=c, the equality holds.This is the case of an equilateral triangle.

So R=2r