 Hi and welcome to the session. Let's work out the following question. The question says three forces p, q and r acting at a point r in equilibrium, the angle between p and q is double the angle between p and r. Prove that r square is equal to q into q minus p. Let us start with the solution to this question. Let the angle between p and r be alpha then angle between and q will be 2 alpha, angle between q and r will be 360 degree minus alpha minus 2 alpha that is equal to 360 degree minus 3 alpha. So, by Lamy's theorem we will have p upon sin 360 degree minus 3 alpha is equal to q upon sin alpha is equal to r upon sin 2 alpha because 360 degree minus 3 alpha is the angle between q and r, alpha is the angle between p and r and 2 alpha is the angle between p and q. This implies r divided by minus sin 3 alpha is equal to q upon sin alpha is equal to r upon 2 sin alpha cos alpha this implies this is p here. So, this implies p upon minus 3 sin alpha minus 4 sin q alpha is equal to q upon sin alpha is equal to r upon 2 sin alpha cos alpha this implies p upon 4 sin square alpha minus 3 is equal to q upon 1 is equal to r upon 2 cos alpha. Therefore, 4 sin square alpha minus 3 is equal to p upon q and cos alpha is equal to r upon 2 q this implies 4 into 1 minus cos square alpha minus 3 is equal to p upon q and cos alpha is equal to r upon 2 q. Now we put the value of cos alpha here and we get 4 into 1 minus r square upon 4 q square minus 3 is equal to p upon q this implies 1 minus r square upon 4 q square sorry here we will have just r square upon q square is equal to p upon q this implies r square upon q square is equal to 1 minus p upon q this implies r square is equal to q square into 1 minus p upon q that is equal to q square into q minus p divided by q. Now q gets cancelled with this q and we have r square is equal to q into q minus p. Now this is what we were supposed to prove in this question I hope that you understood the solution and enjoyed the presentation. Have a good day.