Question

BL and CM are the medians of a ΔABC, right angled at A.Prove that 4(BL square+CM square)=5BC square

Answer 1

Draw the diagram with AC horizontal, AB vertical at 90° to AC.  M is the midpoint of AB and L is the mid point of AC.     ∠A = 90°

IN ΔABL,  BL² = BA² + AL²        Pythagoras theorem
In ΔMAC,  CM² = MA² + AC²            Pythagoras th.

Add the two equations:  
BL² + CM² = (BA² + AC²) + MA² + AL²
                  = BC² + (BA² + AC² )/2²
                  = BC² + BC²/4

Hence, 4 ( BL² + CM²) = 5 BC²