Quandaries
and Queries 

Good morning, I am a student trying to solve math problem. I'd like to calculate the radius of the circle that exactly fits any three points. If the points are (X1,Y1), (X2,Y2), and (X3,Y3), what is the radius of the circle that contains those three points? Thank you very much, Jim 

Hi Jim, The standard construction is illustrated in the diagram.
If P_{1}, P_{2} and P_{3} are distinct points on the circle, L_{1} is the right bisector of the line segment from P_{1} to P_{2} and L_{2} is the right bisector of the line segment from P_{2} to P_{3} then L_{1} and L_{2} meet at the center C. If each P_{i} has coordinates (x_{i}, y_{i}) and M is the midpoint of the line segment from P_{1} to P_{2} then M has coordinates ((x_{1}+x_{2})/2, (y_{1}+y_{2})/2). The slope of the line L_{1} is the negative reciprocal of the slope of the line through P_{1} and P_{2} and thus the equation of L_{1} is
In a similar way find the equation of L_{2} and then solve this pair of equations to find the coordinates of C. The radius of the circle is then the distance from C to P_{1}. Penny 

