How to Construct a Circle From 3 Points

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Manipulative 1: How to construct a circle from 3 non-collinear points. Created with GeoGebra.

A circle can be constructed from 3 non-collinear points.

StepConstructionJustification
1Construct lines segments AB and BC.Euclid. Elements. Book 1. Proposition 1. To draw a straight line from any point to any point.
2Construct the perpendicular bisectors of line segments AB and BC.Euclid. Elements. Book 1. Proposition 10. To bisect a given finite straight line.
3Mark the intersection of the two perpendicular bisectors as M.
4Construct a circle with center at M and radius MA.Euclid. Elements. Book 1. Postulate 3. To describe a circle with any center and radius.

Proof of Construction

StepFigureJustification
1Construction of a circle from 3 points.Start with the completed construction. I say that the circle ABC is a circle that passes through points A, B, and C.
2Construction of a circle from 3 points.By the definition of a perpendicular bisector, the perpendicular bisectors PM and QM divide segments AB and BC into two equal parts and are perpendicular to AB and BC.
3Construction of a circle from 3 points.Construct segments AM, BM, and CM. See Euclid. Elements. Book 1. Proposition 1. To draw a straight line from any point to any point.
4Construction of a circle from 3 points.Since AP=BP AND ∠APM=∠BPM, and PM is in common, by angle-side-angle congruence, ΔAPM=ΔBPM.
5Construction of a circle from 3 points.Since ΔAPM=ΔBPM, line segment AM=BM.
6Construction of a circle from 3 points.Since BQ=CQ AND ∠BQM=∠CQM, and QM is in common, by angle-side-angle congruence, ΔBQM=ΔCQM.
7Construction of a circle from 3 points.Since ΔBQM=ΔCQM, line segment BM=CM.
8Construction of a circle from 3 points.By common notion 1, since AM=BM and BM=CM, then AM=CM. Since AM=BM=CM, points A, B, and C are equidistant from M. So a circle with center at point M and radius AM passes through B and C. Q.E.D..
Table 2: Proof of construction.

Cite this article as:


How to Construct a Circle From 3 Points. 2009-04-17. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/h/htconstructcirclefrm3pnts.html.

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Revision History


2009-04-17: Initial version (McAdams, David.)

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