Question

Prove that in any triangle, the sum of the square of any two sides is equal to twice the square of half of the third side together with twice the square of the median which bisects the third side. (appollonius theorem)

Answer 1

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.