Question

PQR is a right angled triangle having angle Q=90.If QS=SR where S is a point on PR, show that PR^2 =4PS^2-3 PQ^2.

Answer 1

I think what is to be proved is   PR^2 = 4 PS^2.
That is   PR = 2 PS    or,   S is the midpoint of PR.

Draw a perpendicular from S on to QR and QP meeting them at T & U respectively. let angle PSQ = x.  Then angle QSR = 180 - x.
Since in triangle QSR , QS = SR, it is isosceles.  
So angle TSR = (180-x)/2 = 90 - x/2.   Then angle R = x/2.  
So  angle P = 90 - x/2.    That means angle PSU = x/2.  

So then angle USQ = x - x/2 = x/2.
So then the triangle PQS is also isosceles triangle.
So then  QS = PS.    So  PS = QS = SR 
SO   PR^2 = 4 PS^2