Let P be a point in triangle ABC such that angles APB, BPC, and CPA are all 120 degrees. Can lines AB, AC, BC, PA, PB, and PC all have integral lengths?

(In reply to re(2): on another look by Brian Smith)

Sorry Sir,but i never assumed  that for D to be a square that it must be 0.In fact there is no question of D(Discriminant) being a perfect square.Since we have the equation

c^2=x^2 + y^2 + xy=F(x,y)

Actually earlier we assumed x,y,z to be different integers & even i prooved in first comment that they must be diffrent.Now we seek to proove that according to above eqn for some set of integers (x,y) c can be an integer,if we could do so for this particular eqn we can do it for other

So for c to to be an integer F must be a perfect square & we can be assured that F will be integer as x,y are integer, now it all depends on sqrt of F,this will decide whether c is integer or not

And we know for any quadratic eqn to be perfect sqr it must have repeated root,and hence discriminant must be =0

"D=-3*y^2 + 4*c^2=0"
Finally we disern that at a time all can't be integers as in previous discussion

Posted by Nishant on 2007-04-13 15:04:17