Question

prove that the semi perimeter of triangle is less than the sum of its median.

Answer 1

MM is the mid point of ABAB and mcmc is the medians coming from CC. We also have that AB=cAB=c, BC=aBC=a and AC=bAC=b.

Then in the triangle CMACMA we have, by triangle inequality

b<mc+c2(1)b<mc+c2(1)

and in the triangle CMBCMB we have, by triangle inequality

a<mc+c2(2)a<mc+c2(2)

so, (1)+(2)(1)+(2) give us,

a+b<2mc+c(3)a+b<2mc+c(3)

similarily, we have

a+c<2mb+b(4)b+c<2ma+a(5)a+c<2mb+b(4)b+c<2ma+a(5)

now (3)+(4)+(5)(3)+(4)+(5) we get

ma+mb+mc>a+b+c2

Answer 2

M is the mid point of AB and Mc is the median coming from C.We also have AB = c, BC =a, AC =b.

In the triangle CMA we have ,by triangle inequality,

b is lesser than Mc +c/2                                        (1)

In triangle CMB ,

a is lesser than Mc+c/2                                        (2)

(1) +(2) gives,

a+b is lesser than 2Mc+c                                     (3)

Similarly we have ,

a +c is lesse than 2Mb+b                                     (4)

b+c is lesser than 2Ma + a                                   (5)

Adding 3 ,4,5 ; we get ,

Ma+Mb+Mc is lesser than a+b+c/2