A unique circle passes through 3 non-colinear points

As long as the three points are not all on one straight line, there is exactly one circle that passes through them.

Why it’s unique

  1. Perpendicular-bisector construction

    • Draw the perpendicular bisector of the segment joining points A and B.

    • Draw the perpendicular bisector of the segment joining points B and C.

    • Because A, B, C are not collinear, those two bisectors intersect at a single point O.

    • O is equidistant from A, B, and C (that’s how perpendicular bisectors work), so OA = OB = OC.

    • Taking O as the center and OA (or OB or OC) as the radius gives a circle that goes through all three points.

  2. Uniqueness follows from basic geometry
    If some other circle also passed through A, B, and C, its center would have to be equidistant from those three points too, so it would lie at the same intersection of the two bisectors – exactly at O. Only one intersection means only one possible center, hence only one circle.

Edge cases

  • Collinear points: If A, B, C lie on the same straight line, the perpendicular bisectors are parallel and never meet, so no finite-radius circle exists (you could think of the “circle” as a line of infinite radius, but that’s not a true circle in Euclidean geometry).

  • Nearly collinear points: The closer the points are to being collinear, the larger the circle’s radius becomes, but it is still unique unless they are perfectly collinear.

So the familiar rule of thumb is:

Three non-collinear points determine one and only one circle.


“Unique” here means exactly one geometric figure—one specific set of points in the plane—qualifies as the circle through the three given non-collinear points. Concretely:

Aspect What “unique” rules out
Center No two different centers satisfy the equal-distance condition; the perpendicular bisectors meet in one point OO.
Radius Because the center is fixed, the distance OA=OB=OCOA = OB = OC is fixed, so no alternative radius works.
Entire locus A circle is the collection of all points at that fixed distance from OO; with both center and radius nailed down, the locus itself is fixed—there isn’t a second, “different” circle containing the same three points.

So, in the Euclidean plane:

Three non-collinear points    ⟹    one and only one ordered triple (a,b,r)\text{Three non-collinear points} \;\;\Longrightarrow\;\; \text{one and only one ordered triple }(a,b,r)

that satisfies

(x−a)2+(y−b)2=r2for each of the three points.(x-a)^2 + (y-b)^2 = r^2 \quad\text{for each of the three points.}

Any transformation that looks like another circle through the points—say, translating or rotating the whole picture—doesn’t create a new circle; it just re-labels the same set of points in the plane.

What “unique” does not mean

  • Up to similarity or congruence: We aren’t saying “all such circles are congruent”; we are saying there is literally only one candidate.

  • Independent of the plane: If you placed the same three points in a different geometric setting (e.g., on a sphere or in taxicab geometry), the statement could fail or need modification.

  • Including degenerate cases: If the points are collinear, no finite-radius circle exists, so “unique” isn’t even on the table.

In short, “unique” means one and only one circle—specific center, specific radius, specific point set—fits the bill in ordinary Euclidean geometry. 

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