# centre point of a circle given three points

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I need help to calculate the centre point of a circle given three points around the circumference of the circle. (x1,y1),(x2,y2),(x3,y3) I know you have to find the equation of the perpendicular bisector of (x1,y1) to (x2,y2), you know that it passes through the midpoint of the segment: ((x1+x2)/2,(y1+y2)/2), and if the slope of the line connecting (x1,y1) to (x2,y2) is m, the slope of the perpendicular bisector is -1/m. its Working out the equations for the two lines, and finding their intersection, to get the coordinates of the center. Its the algorithm for this (e.g how to implement the code for) that I am not sure of. The first bit is simple addition and division. thanks for looking

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How about this - the distance from the center is the same for all points on the circumference.

So you've got 2 variables (Cx,Cy), and 3 equations:

||C-P1|| == ||C-P2||
||C-P1|| == ||C-P3||
||C-P2|| == ||C-P3||

This should be enough to figure out the variables.

Also you should take the distSQ to get rid of the sqrt and it could be done for 3D too since you have 3 equations.

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C code would look something like this:

////     | x1 + x2 |     | y2 - y1 |   | xc |// 1/2 |         | + s |         | = |    |//     | y1 + y2 |     | x1 - x2 |   | yc |//double s = 0.5*((x2 - x3)*(x1 - x3) - (y2 - y3)*(y3 - y1));double sUnder = (x1 - x2)*(y3 - y1) - (y2 - y1)*(x1 - x3);if(sUnder == 0)	return 0; //insufficient data to calculate centers /= sUnder;double xc = 0.5*(x1 + x2) + s*(y2 - y1); // center x coordinatedouble yc = 0.5*(y1 + y2) + s*(x1 - x2); // center y coordinate

This should work, if I didnt do any mistakes. Maybe there are some ways to optimize it some though.

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I tried that equation for values (x,y)
( 50,50,);
( 400,10);
( 250,500);
and ended up with a result like y = -336953325
???

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When I put in those values I get (414, 326) as the center, which is correct. I made a little win32 app to check it, here is the code:

case WM_PAINT:	{		hdc = BeginPaint(hWnd, &ps);		double x1 = 50;		double y1 = 50;		double x2 = 400;		double y2 = 10;		double x3 = 250;		double y3 = 500;		mark(hdc, (int)x1, (int)y1, 0x000000ff);		mark(hdc, (int)x2, (int)y2, 0x0000ff00);		mark(hdc, (int)x3, (int)y3, 0x00ff0000);		double s = 0.5*((x2 - x3)*(x1 - x3) - (y2 - y3)*(y3 - y1));		double sUnder = (x1 - x2)*(y3 - y1) - (y2 - y1)*(x1 - x3);		if(sUnder != 0)		{			s /= sUnder;			double xc = 0.5*(x1 + x2) + s*(y2 - y1); // center x coordinate			double yc = 0.5*(y1 + y2) + s*(x1 - x2); // center y coordinate			mark(hdc, (int)xc, (int)yc, 0x00000000);		}		EndPaint(hWnd, &ps);	}	break;

Since you got a really big number, so I tink maybe you forgot to divide by sUnder.

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Rutin
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