Construct a Circle Circumscribed about a Triangle MathBitsNotebook.com Topical Outline | Geometry Outline | MathBits' Teacher Resources Terms of Use   Contact Person: Donna Roberts

Remember -- use your compass and straightedge only!

Intuitively, the word "circumscribed", in geometry, refers to drawing a shape around
the outside of another shape so the interior shape "fits snugly" inside of the outside shape.

A circumscribed circle about a triangle is a circle around the outside of a triangle, such that it touches the vertices of the triangle (but never crosses to the interior of the triangle. You may see a "circmscribed circle" referred to as a circumcircle.

circumscribe is like circumference
("circum" = "around" the outside)

See the discussion on "circumcenters" at Circumcenter - Concurrent Bisectors.

Given: an acute scalene triangle ΔABC Construct: Circumscribe a circle about the triangle. Construct a circle about the outside of the triangle touching only the vertices of the triangle.

This construction will focus on the perpendicular bisectors of the sides of the triangle. Remember the theorem: "Any point on a perpendicular bisector of a segment is equidistant from the ends of the segment." Where the perpendicular bisectors intersect (say point O), will be used to establish that OA = OB = OC as the radii of a circle around the outside of the triangle.

Be careful with this construction. Accuracy is very important. If you want your circle to touch all three vertices of the triangle, you will need to be as accurate as possible with each step in this construction. Remember, the circle should just touch each of the vertices. • Does the point of intersection (point O) always lie "INSIDE" the triangle? NOPE! In an acute triangle, it lies "inside" the triangle. In an obtuse triangle, it lies "outside"" the triangle. In an right triangle, it lies "on" the triangle. See more about these locations at Circumcenter - Concurrent Bisectors.

• Why is it sufficient to only construct the perpendicular bisectors of TWO sides of the triangle, instead of all three sides?

Only two perpendicular bisectors will be needed, since when OA = OC and OB = OC, the substitution (or transitive) property gives us that OA = OB, and we have all three radii equal in length, OA = OB = OC. It is not necessary to have the third perpendicular bisector, since we already know mathematically that OA = OB. Yes, you can construct the third perpendicular bisector, if you wish, to verify the location of point O.

Proof of Construction: Draw radii from O to C, from O to A and from O to B. Since point O lies on the perpendicular bisector of , point O is equidistant from A and from B (OA = OB). Any point on a perpendicular bisector of a segment is equidistant from the ends of the segment. Since O also lies on the perpendicular bisector of , point O is also equidistant from B and from C (OB = OC). So, we have OA = OB = OC.
Therefore, point O is at the center of the circle, and the radius of the circle is length OA, and it passes through points A, B and C, the vertices of ΔABC. This gives us circle O circumscribed about ΔABC.