The following sequence of problems provides a proof of the validity of the Simson Line for...

The following sequence of problems provides a proof of the validity of the Simson Line for a triangle. Each problem independently contains a useful, signifi-t cant geometric result.

Formula for the Sides of a Pedal Triangle Suppose that PA = x, PB = y, and PC = z, and that the pedal triangle of ΔABC with respect to P is ΔDEF. where D, E, and F are the feet of the perpendiculars from P to sides , , and , respectively. Then if R is the circumradius of ΔABC, prove that

(Hint: Explain why a circle must pass through A, E, P, and F, then use the result of Problem 8 on the circumcircle of ΔAFE, having circumradius AP = x. What does sin A equal?)