In triangle PQR; QR = p, PR = q and PQ = r with Angle QPR = 2* Angle PQR. The length of all the three sides of the triangle are different.

Is it always true that: p² = q (q + r)?.

Apply sin rule in triangle
p/sin2a = q/sina = r/sin3a (since sin(180-a) = sina)
p/q=sin2a/sina=2cosa
r/q=sin3a/sina=3-4sin^2a(sin^2a=1-cos^2a)
r/q=3-4(1-(p/2q)^2)=3-4+p^2/q^2
r/q=-1+(p/q)^2
p^2=q^2(r/q + 1)=q(r+q)
So, it will be satisfied, but we have to consider all sides are different
180-3a shouldn't be equal to a or 2a which says that 
a shouldn't be 36 or 45 degrees

Posted by Praneeth Yalavarthi on 2007-07-09 12:20:12