It's been a while, and I'm getting nowhere fast with this problem. Perhaps someone could lend me a hand, or at least point me in the right direction. This problem comes from the New York State Regents Exam in Solid Geometry, June, 1950, Part II, Question 22.
Triangle ABC has a right angle at C. Line segment AD is drawn perpendicular to plane ABC. Points E and F are the midpoints of line segments DC and DB respectively. Prove that line EF is perpendicular to the plane ADC.