Right Angle in a Configuration:
Triangle with Hexagon and Equilateral Triangle on Two Sides

  Hint 1: Use a GSP construction and move point A to various positions. Note that A will not always be located on the same side of segment PQ.

  Hint 2:  You will probably need some auxiliary constructions.    Try rotating the figure about  R.

  We need a figure which guarantees a right angle.  For example, can we show P, Q, and R are points on a circle with PQ as a diameter?
  Another example, can we develop a figure where the two segmes PR and RQ must be perpendicular?

  GSP file (or construct your own).