SOLUTION: ABC is a triangel. N is a point inside the triangel. Prove that: AN+BN+CN > 1/2 (AB+BC+AC)

Algebra ->  Triangles -> SOLUTION: ABC is a triangel. N is a point inside the triangel. Prove that: AN+BN+CN > 1/2 (AB+BC+AC)      Log On


   

Question 806516: ABC is a triangel. N is a point inside the triangel. Prove that:
AN+BN+CN > 1/2 (AB+BC+AC)
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
In any triangle, sum of any two sides is always greater than the third.

Consider the triangle ABN:
AN + BN > AB _______________ (1)

Consider the triangle BCN:
BN + CN > BC _______________ (2)

Consider the triangle ACN:
AN + CN > AC _______________ (3)

Adding both sides of (1), (2) and (3) we have
AN + BN + BN + CN + CN + AN > AB + BC + AC
2(AN + BN + CN) > AB + BC + AC
AN + BN + CN > (AB + BC + AC)/2

Hence proved