If #r_1=r_2+r_3+r# prove that the triangle is right angled.?

1 Answer
Jan 16, 2018

We know

#r_1=Delta/(s-a)#

#r_2=Delta/(s-b)#

#r_3=Delta/(s-c)#

#r=Delta/s#

Inserting these values in the given relation.

#r_1=r_2+r_3+r#

#=>r_1-r=r_2+r_3#

#=>Delta/(s-a)-Delta/s=Delta/(s-b)+Delta/(s-c)#

#=>1/(s-a)-1/s=1/(s-b)+1/(s-c)#

#=>2/(2s-2a)-2/(2s)=2/(2s-2b)+2/(2s-2c)#

#=>1/(b+c-a)-1/(b+c+a)=1/(c+a-b)+1/(a+b-c)#

#=>((b+c+a)-(b+c-a))/((b+c)^2-a^2)=(c+a-b+a+b-c)/(a^2-(b-c)^2)#

#=>(2a)/((b+c)^2-a^2)=(2a)/(a^2-(b-c)^2)#

#=>(b+c)^2-a^2=a^2-(b-c)^2#

#=>(b+c)^2+(b-c)^2=a^2+a^2#

#=>2b^2+2c^2=2a^2#

#=>b^2+c^2=a^2#

This means the triangle is right angled.