Given three points, it is possible to draw a circle that passes through all three*. This page shows how to construct (draw) a circle through 3 given points with compass and straightedge or ruler. It works by joining two pairs of points to create two chords. The perpendicular bisectors of a chords always passes through the center of the circle. By this method we find the center and can then draw the circle.
This is virtually the same as constructing the circumcircle a triangle. If you draw three lines linking the given points, you will get a triangle. The circumcircle passes through all three vertices, just as here.
* What if the points are collinear?
If the three points are collinear (all lying on a straight line) then the circle passing through all three will have a radius of infinity. So there is no practical circle that can pass through three collinear points. If you tried the construction, you would find that the two radii (JO, LO) would be parallel and so never meet at a center. In the language of mathematics, we might say they do intersect, but at infinity.
Printable step-by-step instructions
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
Proof
The image below is the final drawing above with the red items added.
| Argument | Reason | |
|---|---|---|
| 1 | JK is the perpendicular bisector of AB. | By construction. For proof see Constructing the perpendicular bisector of a line segment |
| 2 | Circles exist whose center lies on the line JK and of which AB is a chord. (* see note below) | The perpendicular bisector of a chord always passes through the circle's center. |
| 3 | LM is the perpendicular bisector of BC. | By construction. For proof see Constructing the perpendicular bisector of a line segment |
| 4 | Circles exist whose center lies on the line LM and of which BC is a chord. (* see note below) | The perpendicular bisector of a chord always passes through the circle's center. |
| 5 | The point O is the center of the only circle that passes through A,B,C. | O is the only point that lies on both JK and LM, and so satisfies both 2 and 4 above. |
- Q.E.D
* Note
Depending where the center point lies on the bisector, there is an infinite number of circles that can satisfy this.
Two of them are shown below.
Steps 2 and 4 work together to reduce the possible number to just one.
Try it yourself
Click here for a printable worksheet containing two problems that use this construction technique. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.Other constructions pages on this site
Lines
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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