Problem: Let be the point of intersection of the diagonals of convex quadrilateral , and let and be the centers of the circles circumscribing triangles and , respectively. Then
Answer Choices:
A. is a parallelogram
B. is a parallelogram if and only if is a rhombus
C. is a parallelogram if and only if is a rectangle
D. is a parallelogram if and only if is a parallelogram
E. none of the above are true
Solution:
The center of a circle circumscribing a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle. Therefore, , and are the intersections of the perpendicular bisectors of line seg. ments and . Since line segments perpendicular to the same line are parallel (and since lines and coincide and lines and coincide), is a parallelogram.