Median is any line in a triangle which divides the given triangle segment in two equal parts.
Let us first consider a median from point A dividing the line BC.
D is any point on the segment BC
We know one basic fact about triangles that the sum of any two sides of a triangle is always greater than twice the median which is dividing the third side in two equal halves.
AB + AC > 2 × AM ……………. (I)
Now let us consider median from point B dividing the segment AC.
Similarly we can write here as follows:
AB + BC > 2 × BN …………… (II)
Now let us consider median from point C dividing the segment AB.
AC + BC > 2 × OC …………….. (III)
Now adding equations (I), (II) and (III)
AB + AC + BC + AB + AC + BC > (2AM + 2BN + 2OC)
2AB + 2BC + 2AC > 2 × (AM + BN + OC)
2 × (AB + AC + BC) > 2 × (AM + BN + OC)
AB + AC + BC > AM + BN + OC
Hence it is proved that the perimeter of a triangle is more than the sum of length of the medians.
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