CRUX 1983 . Proposed by K.R.S. Sastry, Dodballapur, India. A convex quadrilateral ABCD has an inscribed circle with center I and also has a circumscribed circle. Let the line parallel to AB through I meet AD in A' and BC in B'. CRUX 1991 .
M be the foot of the perpendicular from O to a fixed line , and let P be a variable point on .
Let be the circle with diameter PM intersecting and again at X and Y, respectively. Prove that the line XY always passes through a fixed point. CRUX 1998 . Proposed by John Clyde, student, New Plymouth High School, New Plymouth, Idaho. CRUX 2444 .