in a trainagle,R & r are the circumradius and inradius respectively then the H.M of the exradius the triangle is ???????????

Dear Zyesha,

 let r1 , r2 , r3 be the three exradii 

r1= Δ/(s-a)   ,r2= Δ/(s-b) , r3= Δ/(s-c)---------------1

where  Δ is the area of triangle   and  s = (a+b+c)/2

the harmonic means of the three exradii are

H.M = 3/(1/r1 + 1/r2 + 1/r3) ---------------2

putting the value of r1,r2,r3 from  1  to 2

 H.M= 3Δ/(s-a+s-b+s-c)  

so H.M = 3Δ/s

and we know that

r = Δ/s

so H.M of exradii = 3 r

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