section_path: Analytic results/Equations/Cartesian coordinates/3-point form
section_name: 3-point form
paragraph_text: 3-point form
  
  The equation of the circle determined by three points (x 1, y 1), (x 2, y 2), (x 3, y 3) {\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})} not on a line is obtained by a conversion of the 3-point form of a circle equation: (x − x 1) (x − x 2) + (y − y 1) (y − y 2) (y − y 1) (x − x 2) − (y − y 2) (x − x 1) = (x 3 − x 1) (x 3 − x 2) + (y 3 − y 1) (y 3 − y 2) (y 3 − y 1) (x 3 − x 2) − (y 3 − y 2) (x 3 − x 1). {\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.}