If P₁, P₂ and P₃ are the lengths of the altitudes drawn from the vertices A, B and C of A B C respectively, then A P₁ + B P₂ + C P₃ =

Step-by-step solution

We know that: P₁= 2 a , P₂= 2 b , P₃= 2 c Now, A P₁ + B P₂ + C P₃ aligned & = a A 2 + b B 2 + c C 2 & = 2 R A A+2 R B B+2 R C C 2 & = R( 2 A+ 2 B+ 2 C) 2 & = 2 R( (A+B) (A-B)+ C C) 2 & = 2 R( C (A-B)- C (A+B)) 2 & = 4 R C[ B C] 2 & = 4 R ( a 2 R ) ( b 2 R ) ( c 2 R ) 2 = 1 R^2 a b c 4 & = 1 R^2 R= 1 R aligned