Centre of a circle

Question

I was wondering whether if given any three point ( all distinct from each other) on the circumference of circle can I always determine the centre of the a circle. If not what scenarios would this not be applicable.

Answer 1

Yes, if the three points are indeed on a circle (i.e. not aligned) and are distinct, you can always retrieve the center.

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The center is the intersection of the bisectors of the points, in pairs. The bisector of two distinct points can always be constructed, and the bisectors can only be parallel if the points are aligned.

Answer 2

Yes of course, three (not aligned) points determine a circle and then we can always find its center.

See for example

• Equation of circle passing through 3 given points

and once we have the equation in the form $x^2+y^2+ax+by+c=0$, we can determine the center completing the square and reducing to the form $(x-x_C)^2+(y-y_C)^2=R^2$.

Refer also to the related

• Get the equation of a circle when given 3 points

Answer 3

Two points $A$, $B$ on a circle determine the chord $AB$. The perpendicular bisector of $AB$ goes through the centre of the circle. Having a third point on the circle gives you another chord, say $AC$ or $BC$. The perpendicular bisector of this second chord also goes through the centre of the circle. It follows that the centre is the point of intersection of those two perpendicular bisectors.