Choose any four points in a plane, such that no three are collinear and the four do not lie on a circle.

Show that one of the points must lie within the circle formed by the other three.

If this were not the case, then the quadrilateral formed by the four points would always have, for any three points chosen to define the circle, the fourth point outside the circle.

Erect a radius of the circle as a perpendicular bisector of one of the sides of the quadrilateral that are chords of the circle.  A different circle can be made by moving the center backward, away from the chord that it bisects, increasing the radius to be large enough at some point to cause the point formerly outside the circle to be on the circle.  But at the same time, the distances from the center on the opposite side also increase so that the point formerly on the circle is now inside the circle.  So some point can indeed be made to lie inside such a circle.

Edited on February 19, 2004, 2:19 pm

Posted by Charlie on 2004-02-19 14:17:56