ABC is a triangle where the respective lengths of the sides BC, AC and AB are denoted by X, Y and Z. It is known that 3*∠ BAC + 2*∠ ABC = 180^o

Prove that: X² + Y*Z = Z²

Let D be a point on AB such that AD = AC, and let O be the centre of the circle through A, C and D.

/ BCO    = C - / ACO
            = C - A/2
            = (180 - A - B) - A/2
            = 90                 (using the fact that 3A + 2B = 180)

So BC is a tangent to the circle at C, and the intersecting tangent/chord theorem

gives:               X² = Z(Z - Y)                 i.e.        X² + YZ = Z²

Posted by Harry on 2011-02-15 17:27:41