Question

show that the sum of the three median of a triangle is less than its perimeter

Answer 1

Let ABC be the triangle and D. E and F are midpoints of BC, CA and AB respectively.Recall that the sum of two sides of a triangle is greater than twice the median bisecting the third side, Hence in ΔABD, AD is a median ⇒ AB + AC > 2(AD)Similarly, we get BC + AC > 2CF BC + AB > 2BE On adding the above inequations, we get (AB + AC) + (BC + AC) + (BC + AB )> 2AD + 2CD + 2BE 2(AB + BC + AC) > 2(AD + BE + CF) ∴ AB + BC + AC > AD + BE + CFHence, we can say that the perimeter of a triangle is greater than the sum of the medians.

Answer 2

refer to the attachment