SOLUTION: ABC is a triangel. N is a point inside the triangel. Prove that: AN+BN+CN

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

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) (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
RELATED QUESTIONS
I have a problem as follows : Point M and N are taken on sides AB & AC of the triangle (answered by KMST)

there is a triangel ABC which AC lenght is 10 cm and lenght AB is 15 cm and lenght BC is... (answered by richwmiller)

ABC is a triangle. D is a point on AB such that AD = 1/4 AB and E is a point on AC such... (answered by Theo)

ABC is an obtuse - angled triangle at A, AB = AC A=and intersects BC line segment at D (answered by Edwin McCravy)

The circumcenter of triangle ABC is S. AS extended intersect BC at M. BS extended... (answered by rothauserc,ikleyn)

In ΔABC, m∠A = 90� and AN is an altitude. If AB = 20 in and AC = 15 in, find... (answered by josgarithmetic,greenestamps)

A (4,0) B (0,8) C (7,4). A point N lies on AB as such so that CN is perpendicular to AB.... (answered by josgarithmetic,stanbon,josmiceli,MathTherapy)

In an isosceles triangle labelled ABC, with BC as the base, and AB as the hypotenuse, and (answered by greenestamps)

In a triangle ABC, angle A = 2 angle B. How would you prove that BC� = AC� + AB �... (answered by richard1234)