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The image shows a triangle.
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.
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.
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 .