Imagine a point placed anywhere on the perpendicular bisector you constructed. How would the distance from to compare to the distance from to ? Explain your reasoning.
Each student in your group should choose one triangle. It’s okay for two students to choose the same triangle as long as all three triangles are chosen by at least one student.
Construct the circumscribed circle of your triangle.
After you finish, compare your results. What do you notice about the location of the circumcenter in each triangle?
Student Lesson Summary
We saw that some quadrilaterals have circumscribed circles. Is the same true for triangles? In fact, all triangles have circumscribed circles. The key fact is that all points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment.
Suppose we have triangle and we construct the perpendicular bisectors of all three sides. These perpendicular bisectors will all meet at a single point called the circumcenter of the triangle (label it ). This point is on the perpendicular bisector of , so it’s equidistant from and . It’s also on the perpendicular bisector of , so it’s equidistant from and . So it is actually the same distance from and. We can draw a circle centered at with radius . The circle will pass through and too because the distances and are the same as the radius of the circle.
In this case, the circumcenter happens to fall inside triangle , but that will not always be the case. The images show cases where the circumcenter is inside a triangle, outside a triangle, and on one of the sides of a triangle.
Glossary
circumcenter
The circumcenter of a triangle is the intersection point of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle.
In this diagram, the circumcenter is point .
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Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.