Problem 3 of the Canadian Mathematical Olympiad 1991

Let C be a circle and P a given point in the plane. Each line through P which intersects C determines a chord of C. Show that the midpoints of these chords lie on a circle.

Solution by Steve Dinh, a.k.a. Vo Duc Dien (dedicated to Minh Dao)

Itís easily recognized that such a circle needs to pass through the center O of circle C. Draw an arbitrary secant through P that cuts chord AB on C. Let E be the midpoint AB. Since OA = OB = radius OE ⊥ AP, E lies on a circle having OP as a diameter.

The above process applies to any chord initiating from P and intersecting circle C. Therefore, all the midpoints lie on the circle with diameter OP.