Equation of a Circle with Three Points: Guide and Explanation

[abia ubong]

hi,
my friend once asked me if i could find the equation of a circle,when given three points,i mean (x,y) (x2,y2) (x3,y3).please let me know if it is possible.

 

[inha]

I think you can. All pairs (xi,yi) should fullfill xi^2+yi^2=r^2 where r is the radius. You should be able to construct a linear set of equations and solve for r.

 

[uart]

Yes it is possible. Three points will always uniquely specify a circle providing that they are not co-linear.

 

[snoble]

It definitely is possible. What you're doing is finding a point that is equidistant from all three points and that will give you the centre of the circle. Then the radius is just the distance from the centre to any point. If you ever watch cop shows and they are talking about triangulating a radio signal from 3 points this is what they are referring to (sort of).

So how do you do it? It's actually really easy when you think of it this way. If you take two random points and graph the points that are equidistant from both you get a line. Specifically a line that intersects the line segment between the two points at 90 degrees at the half way point. Just pick two pairs of points (one point will be in both pairs). Draw the equidistant lines for both pairs. The intersection point is the point of equidistance from all 3. Hence it is the centre of the circle.

 

[abia ubong]

thanks a lot

 

[dextercioby]

Proof:3 noncolinear points determine a nondegenerate triangle uniquely.Any nondegerate triangle has a circumscribed circle.EndProof.

Daniel.