# Centre of a circle [closed]

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.

## closed as off-topic by user21820, Namaste, Saad, José Carlos Santos, Yves DaoustNov 24 '18 at 23:01

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• The three points have to be all distinct from each other, but I suppose you were assuming that anyway. – David K Nov 24 '18 at 14:26
• @DavidK And also not aligned. – gimusi Nov 24 '18 at 14:27
• Yes you are correct David K sorry about that. – odesinit Nov 24 '18 at 14:27
• I'm bit confused with aligned what does that mean in this context? – odesinit Nov 24 '18 at 14:30
• en.wikipedia.org/wiki/… – Jean-Claude Arbaut Nov 24 '18 at 14:35

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

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.

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.

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

See for example

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

• IMO, this is only a restatement of the question. You should prove that the system of equations has always a unique solution. – Yves Daoust Nov 24 '18 at 14:41
• @YvesDaoust Euclid already proved that for me more that 2000 years ago :) – gimusi Nov 24 '18 at 14:45
• @YvesDaoust Note also that in the link I given there are many different methods which show how to solve the system. – gimusi Nov 24 '18 at 15:08
• @YvesDaoust By the same approaches we can generalize the result. Anyway since the question was on how to find the center, I was focused to give some hint on that and not for a general proof for a well known result by Euclidean geometry. I can ask if the asker is also interested in that proof. – gimusi Nov 24 '18 at 15:23
• @YvesDaoust Note that he is stating "given any three point ( all distinct from each other) on the circumference of circle..." then the existence of the circle is out of the dicussion. That was my interpretation of the OP. I've just asked for a specific clarification about that to the asker. – gimusi Nov 24 '18 at 15:27