How to Draw a Circle Through Any Three Points

A compass and straight edge are all that are required to compose this circle.

Any three non-linear points describe a unique circle. The circle that passes through any three points can be drawn using only a straight edge and compass in a process known as construction. Construction is a technique dating back to the time of the ancient Greek mathematicians and can be used to draw many geometric figures and demonstrate various mathematical concepts.

Things You'll Need

• Pen and paper
• Straight edge
• Compass

Instructions

1. Draw three points in the central area of the paper. The middle point should be higher than the other two.

2. Connect the center point to the other two points using two straight lines.

3. Place the needle of the compass on the leftmost point, and spread the compass to approximately two-thirds the length between the leftmost point and the central point.

4. Draw a short arc with the compass on either side of the line between the leftmost and central points.

5. Without adjusting the compass width, move the needle to the central point and again draw a short arc above and below the same line, so that these arcs cross the arcs previously drawn.

6. Draw a straight line between the intersections of the two arcs above the line and the intersection of the two arcs below the line. This new line is the perpendicular bisector of the line between the leftmost and central points.

7. Place the compass needle on the rightmost point, adjust it to two-thirds the distance between the rightmost and central points, and draw arcs on either side of the line between the rightmost and central points.

8. Move the compass needle to the central point (again without adjusting the compass), and once more draw a short arc above and below the line, so that these arcs cross the arcs just drawn.

9. Draw a straight line between the intersections of the two arcs above the line and the intersection of the two arcs below the line. This new line is the perpendicular bisector of the line between the rightmost and central points.

10. Place the needle of the compass at the point where the two perpendicular bisector lines cross. This is the center of the circle to be drawn.

11. Extend the compass so that its other end reaches one of the three points (all three should be equidistant). Keeping the compass needle on the center point, draw a complete circle that passes through all three points.

Tips & Warnings

• This construction if sometimes called the "circumcircle of a triangle" because rather than using three points, we could instead start with the three vertices of a triangle.

References

• "Three-points Circle": Zef Daman; 2010
• Math Open Reference: Circle Through Three Points: John Page; 2009

Resources

• Northern Virginia Community College: To Construct a Circle Given Three Points