Prove that if three points are not collinear then there exis

Prove that if three points are not collinear then there exists a unique circle that passes through them.

Solution

If the three points are not colinear, you can prove that there exists an unique circle by proving that it\'s center lies on two perpendicular bisectors of the segments connecting the tree points.[The existence is done by proving that the circle with the center at the intersection of two perpendicular bisectors and radius the distance to one of the points works; the uniqueness is done by proving that any circle passing through the three points has that center and radius].

If the three points are co linear, there is no circle (unless you consider a line as a circle of infinite radius). This can be proven by showing that if the three points are on a line in the order A,B,CA,B,C(i.e. BB between AA and CC), then for any point in the plane we have

Added If the three points are not necessarily distinct (which from how the problem is stated I don\'t think is the case), then it is easy to prove there are infinitely many circles.

Case 1: All three points are the same, then it is easy to construct infinitely many circles.

Case 2: Two points are the same, but third is distinct. Ignore one of the two equal points. Then any circle with the center on the Perpendicular Bisector of the segment connecting the two points, and the right radius will do.