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Question

In ABC, the incircle touches the sides BC, CA and AB, respectively, D, E, and F. If the radius of the incircle is 4 units and BD, CE and AF are consecutive integers, then the value of s3, where s is a semi-perimeter of triangle, is 


Solution

 
Let BD=x,CE=x+1 and AF=x+2 Then
CD=CE=x+1=sc
AE=AF=x+2=sa
BF=BD=x=sb
On adding,  we get
3s(a+b+c)=3x+3s=3x+3sa=x+2,sb=x,sc=x+1 
Now, r=s=s(sa)(sb)(sc)
Or,  4=(x+2)x(x+1)3(x+1)
Or, x2+2x=48
x=6
Hence, s=21s3=7 

Mathematics

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