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)?.

From the law of sines,

    q          p           p              p
 -------- = -------- = --------- = ----------------
  sin(Q)     sin(P)     sin(2Q)     2 sin(Q)cos(Q)

                      or
            
 cos(Q) = p/2q
               
From the law of cosines,

 q^2 = p^2 + r^2 - 2pr cos(Q)

     = p^2 + r^2 - 2pr (p/2q)

          or

 p^2(q - r) = q(q + r)(q - r)

If q != r, then p^2 = q(q + r).

If q = r, then Q = R = 45 and P = 90

   and therefore p^2 = q^2 + r^2 = q^2 + qr

                     = q(q + r)

Thus, p^2 = q(q + r) whether q = r or not.

Posted by Bractals on 2007-06-11 12:49:11