Problem: Let be an interior point of circle other than the center of . From all chords of which pass through , and determine their midpoints. The locus of these midpoints is
Answer Choices:
A. a circle with one point deleted
B. a semicircle with one point deleted
C. a circle if the distance from to the center of is less than one half the radius of ; otherwise a circular arc of less than
D. a semicircle
E. a circle
Solution:
Suppose is the given point, is the center of circle , and is the midpoint of a chord passing through . Since is lies on a circle having for the diameter. Conversely, if is any point on the circle with diameter , then the chord of the given circle passing through and (the chord of the given circle tangent to circle at if ) is perpendicular to . Hence is the midpoint of this chord and therefore belongs to the locus.