Question

Sum of squares of medians if sum of square of sifes are given

Answer 1

 Apollonius theorem states that the sum of the squares of two sides of a triangle is equal to twice the square of the median on the third side plus half the square of the third side. Hence AB2 + AC 2 = 2BD 2 + 2AD 2                                  = 2 × (½BC)2 + 2AD2                                  = ½ BC2 + 2AD2 ∴ 2AB2+ 2AC 2 = BC2 + 4AD2  → (1) Similarly, we get 2AB2 + 2BC2 = AC2 + 4BE2   → (2)2BC2 + 2AC2 = AB2 + 4CF2   → (3) Adding (1) (2) and (3), we get 4AB2 + 4BC2 + 4AC 2 = AB2 + BC2 + AC2 + 4AD2 + 4BE2 + 4CF2      3(AB2 + BC2 + AC2) = 4(AD2 + BE2 + CF2)       Hence, three times the sum of squares of the sides of a triangle is equal to four times the sum of squares of the medians of the triangle.