Question

O is any point in the interior of Δ ABC. Prove that

(i) AB + AC > OB + OC

(ii) AB + BC + CA > OA + OB + OC

(iii) OA + OB + OC > $\frac{1}{2}$(AB + BC + CA)

Answer 1

It is given that, is any point in the interior of

We have to prove that

(1) Produced to meet at.

In we have

.........(1)

And in we have

.........(2)

Adding (1) & (2) we get

HenceProved.

(2) We have to prove that

From the first result we have

..........(3)

And

.........(4)

Adding above (4) equation

HenceProved.

(3) We have to prove that

In triangles, and we have

Adding these three results

HenceProved.