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Circumcenter of a Triangle
Geometry construction using a compass and straightedge
This construction assumes you are already familiar with Constructing the Perpendicular Bisector of a Line Segment.
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Your work should look like this |
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We start with a triangle ABC.
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1. Find the bisector
of one of the triangle sides. Any one will do.
See Constructing the Perpendicular Bisector of a Line Segment for detailed instructions.
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| 2. Repeat for the another side. Any one will do. |
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| 3. Mark the point where these two perpendiculars intersect as point O.
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| (Optional step) Repeat for the third side.
This will convince you that the three bisectors do, in fact, intersect at a single point.
But two are enough to find that point. |
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Done The point O is the circumcenter of the triangle ABC.
Note: This point may be outside the triangle. This is normal.
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Note:
The circumcenter is the center of a triangle's circumcircle,
and the construction of the circumcircle is almost the same as this one,
with the addition of the last step to actually draw the circle.
See
Constructing the circumcircle of a triangle.
Proof
The image below is the final drawing above with the red labels added.
- 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 on the right.
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 triangle circumcenter problems.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Constructions pages on this site
Lines
Angles
Triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
Non-Euclidean constructions
(C) 2009 Copyright John Page
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