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Converters Constants Calculator Analytical Geometry
Circle Vectors Straight Line Parabola Ellipse Hyperbola
Coordinate Geometry [Triangle] Coordinate Geometry [Quadrilateral]
Operations:
Intersection of two circles
Intersection of a circle and a straight line
Equation of circumcircle of a triangle
Image of a circle
Equation of incircle of a triangle
Equation of excircles of a triangle
Equation of common chord of two circles
Parametric Equation of Circle
Chord of Contact of Circle

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OPERATIONS:

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Distance between two points
Line joining two points
Line passing through a point having a slope
Division of Points into Certain Ratio
Division of Points by another Point
Division of Straight line
Parametric Equation of Straight Line
Points on a Straight Line at a Distance
Perpendicular Form of Straight Line from General Form
Condition of three lines being concurrent
Distance between two parallel straight lines
Distance between a straight line and a point
Distance between a point from a line parallel or perpendicular to another line
Angle between two straight lines
Image of a point with respect to a line
Image of a line with respect to a line
Bisectors of two straight lines
Intercept of Straight Line divided at a point at a certain ratio
At a specific angle with the X-axis maintaining distance from the origin
Parallel to a Straight line
Perpendicular to a Straight Line
Passing through the Intersection Point of Two Straight Lines
Passing through a point and having a distance from another point
Being parallel to another straight line and having a distance from another point
Being perpendicular to another straight line and having a distance from another point
Creating an angle with another straight line

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Equation of a Circle passing through three points

A circle may pass through three vertices \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\).
Input these data to find the equation of the circle:










Theory [Equation of a Circle passing through three points]

If the circle passes through \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\), then as the equation of a circle is: $$x^2+y^2+2gx+2fy+c=0$$ The three points must satisfy this equation, from which we get: $$x_1^2+y_1^2+2gx_1+2fy_1+c=0$$ $$x_2^2+y_2^2+2gx_2+2fy_2+c=0$$ $$x_3^2+y_3^2+2gx_3+2fy_3+c=0$$ Considering \(p=-(x_1^2+y_1^2)\), \(q=-(x_2^2+y_2^2)\) and \(r=-(x_3^2+y_3^2)\), we can rewrite the equations and subtract first equation from the second and second from the third, to arrive at: $$2g(x_1-x_2)+2f(y_1-y_2)=a-b$$ $$2g(x_2-x_3)+2f(y_2-y_3)=b-c$$ Solving these equations give us \((g,f)\): $$g=\frac{1}{2}\frac{(y_1-y_2)(c-b)+(y_2-y_3)(a-b)}{(x_1-x_2)(y_2-y_3)-(x_2-x_3)(y_1-y_2)}$$ $$f=\frac{1}{2}\frac{(x_2-x_3)(b-a)+(x_1-x_2)(b-c)}{(x_1-x_2)(y_2-y_3)-(x_2-x_3)(y_1-y_2)}$$ Plunging these values in any of the three equations, we can solve for \(c\).

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