Let P,Q,R be three points in the plane which do not all lie on the same line. Then there is a unique circle that passes through all three of them. See Figure 3.20. There are several ways to conï¬rm this assertion.
(a) A general circle has equation
x2 + ax + y2 + by = c.
Thus there are three undetermined parameters. And the three pieces of information provided by the fact that the circle must pass through P =( p1,p2), Q =( q1,q2), R =( r1,r2) (and therefore these three points must satisfy the equation) will determine those parameters. Use this idea to ï¬nd the unique circle that passes through (1,2), (2,3), and (4,9).
(b) There is a well-deï¬ned perpendicular bisector to the segment PQ. This line represents the set of all points that are equidistant from P and Q. There is also a welldeï¬ned perpendicular bisector to the segment QR. This line represents the set of all points that are equidistant from Q and R. The intersection of these two linesâ which will be a single point Câwill be the unique point that is equidistant from all three of P,Q,R. That must be thecenter of the circle. See Figure 3.21. The distance of C to P will be the radius. Use this idea to ï¬nd the unique circle that passes through (1,0), (0,1), (1,1).
Let P,Q,R be three points in the plane which do not all lie on the same line. Then there is a unique circle that passes through all three of them. See Figure 3.20. There are several ways to conï¬rm this assertion.
(a) A general circle has equation
x2 + ax + y2 + by = c.
Thus there are three undetermined parameters. And the three pieces of information provided by the fact that the circle must pass through P =( p1,p2), Q =( q1,q2), R =( r1,r2) (and therefore these three points must satisfy the equation) will determine those parameters. Use this idea to ï¬nd the unique circle that passes through (1,2), (2,3), and (4,9).
(b) There is a well-deï¬ned perpendicular bisector to the segment PQ. This line represents the set of all points that are equidistant from P and Q. There is also a welldeï¬ned perpendicular bisector to the segment QR. This line represents the set of all points that are equidistant from Q and R. The intersection of these two linesâ which will be a single point Câwill be the unique point that is equidistant from all three of P,Q,R. That must be thecenter of the circle. See Figure 3.21. The distance of C to P will be the radius. Use this idea to ï¬nd the unique circle that passes through (1,0), (0,1), (1,1).
