Question

p and q are the midpoints of the sides CA and CB respectively of a triangle ABC right angled at C prove that
4BP square+4BC square+AC- square

Answer 1

In ΔACQ
AQ² = AC² + QC²
AQ² = AC² + (BC/2)²
AQ² = AC² + BC²/4    .............(1)

In ΔBPC
BP² = BC² + PC²
BP² = BC² + (AC/2)²
BP² = BC² + AC²/4        ...........(2)

Adding (1) and (2)
AQ² + BP² = AC² + BC² + AC²/4 + BC²/4 

4(AQ² + BP²) = 5AC² + 5BC²  
4(AQ² + BP²) = 5(AC² + BC²)
4(AQ² + BP²) = 5(AB²)