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)?.

Lets call angle PQR (¥á) and angle QPR (2¥á).  By the law of sines:

p / sin(2¥á) = q / sin(¥á)   or  p / [2*sin(¥á)cos(¥á)] = q / sin(¥á)  

cos(¥á) = p/(2q)

By the law of cosines:

q©÷ = r©÷ + p©÷ - 2pr*cos(¥á)   or     q©÷ = r©÷ + p©÷ - p©÷r/q

q©÷ = r©÷ + p©÷(1 - r/q)

p©÷ = (q©÷ - r©÷) / (1 - r/q)

p©÷ = [(q + r)*(q - r)] / [(q - r)/q]

p©÷ = (q + r) / [1/q]

p©÷ = q(q + r)

By definition, ¥á < 60¨¬.

Posted by hoodat on 2007-06-11 13:48:24