The incircle of a scalene triangle A B C touches B C at D, C A at E and A B at F . Let r_A be the radius of the circle inside A B C which is tangent to and the sides A B and A C . Define r_B and r_C similarly. If r_A=16, r_B=25 and r_C=36 , determine the radius of .

Step-by-step solution

Let inradius of A B C be r In I D C: I C=r cosec C 2 and I₁ C=36 cosec C 2 aligned & I₁=r cosec C 2 -36 cosec C 2 =36+r & r-36 r+36 = C 2 aligned Similarly r-25 r+25 = B 2 and r-16 r+16 = A 2 aligned & A 2 = -(B+C) 2 & A 2 = B 2 C 2 - B 2 C 2 aligned aligned & r-16 r+16 = 1- ( r-25 r+25 )^2 1- ( r-36 r+36 )^2 & - r-25 r+25 r-36 r+36 & r-16 r+16 = 10 r r+25 12 r r+36 - (r-25)(r-36) (r+25)(r+36) & (r-16)(r+25)(r+36)=(r+16)(120 r- & . (r^2-61 r+900 ) ) & r (r^2+61 r+900-181 r+900+r^2 )=16 (r^2+61 r . & .+900+181 r-900