Lesson 7

Circles in Triangles

In this lesson, students apply what they’ve learned about angle bisectors to construct a triangle’s inscribed circle. Then, students use their knowledge of circumcenters and incenters to prove a property of equilateral triangles.

Students use appropriate tools like tracing paper, straightedge, compass, or dynamic geometry software strategically (MP5) when they construct the inscribed circle of an arbitrary triangle. They have an opportunity to construct a viable argument (MP3) when they write a proof that for an equilateral triangle, the incenter and circumcenter coincide.

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Construct the inscribed circle of a triangle.

Let’s construct the largest possible circle inside of a triangle.

Geometry toolkits (HS)

I can construct the inscribed circle of a triangle.

Building On

Addressing

Building Towards

HSG-C.A.3

circumcenter

The circumcenter of a triangle is the intersection of all three perpendicular bisectors of the triangle’s sides. It is the center of the triangle’s circumscribed circle.
circumscribed

We say a polygon is circumscribed by a circle if it fits inside the circle and every vertex of the polygon is on the circle.
cyclic quadrilateral

A quadrilateral whose vertices all lie on the same circle.
incenter

The incenter of a triangle is the intersection of all three of the triangle’s angle bisectors. It is the center of the triangle’s inscribed circle.