Students construct the circumcenter, or the point where the perpendicular bisectors of the sides of a triangle intersect, of the triangle from the previous activity. Students determine that the circumcenter is equidistant from the vertices of the triangle using the properties of perpendicular bisectors. Finally, they construct the circumscribed circle, observing that this is possible because of the equal distances already noted, making use of the structure of a circle (MP7).

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Due to the precision of the tools, students’ circles may not pass through all three vertices. Explain that if we had more precise tools, the distances would be the same and the circle would pass through all the vertices.

The goal is to conclude that all triangles have a circumscribed circle.

• “What kind of circle did you construct in relation to the triangle?” (It is a circumscribed circle.)
• “We saw that only some quadrilaterals have a circumscribed circle. Do all triangles have a circumscribed circle, or only some?” (All triangles have a circumscribed circle, because what we did for triangle could be done for any triangle. We used properties of perpendicular bisectors, not a special property of this triangle.)
• “What other special lines in triangles have we seen that all meet at a central point?” (In the unit “Coordinate Geometry,” we saw that the medians and the altitudes in a triangle also have single points of intersection.)

Tell students that the point where all three perpendicular bisectors of a triangle intersect is called the triangle’s circumcenter. Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

The three perpendicular bisectors of the sides of a triangle meet at a single point, called the triangle’s circumcenter. This point is the center of the triangle’s circumscribed circle. (Theorem)

If time permits, ask: “For the triangle, once we drew two perpendicular bisectors, we knew the third would pass through the point of intersection of the first two. Why doesn’t a similar argument work for a quadrilateral? Use a general quadrilateral as an example.” (Draw the perpendicular bisectors of segments and and call their point of intersection . This point is equidistant from points and . In a triangle, there are only three vertices, so that covers all of them. However, now we have four vertices. We have no way to guarantee that point is equidistant from point .)

In this activity, students compare the circumcenters of acute, obtuse, and right triangles. They find that the circumcenter of a right triangle lies on its hypotenuse, the circumcenter of an acute triangle lies inside the triangle, and the circumcenter of an obtuse triangle lies outside the triangle.

In the digital version of the activity, students use an applet to make conclusions about the circumcenters of acute, obtuse, and right triangles. The applet allows students to move the vertices of a circumscribed triangle and observe how the location of the triangle’s circumcenter changes. 

This activity works best when each student has access to devices that can run the applet because students will benefit from seeing the relationship in a dynamic way.

If students struggle to get started, ask them to look back at their work from the previous activity.