In this activity, students create an arbitrary triangle and use what they know about angle bisectors and constructions to find the incenter. Then they construct segments measuring the distance from the incenter to the sides of the triangle. Finally, students construct the triangle’s inscribed circle, or the circle that is tangent to all three sides of the triangle.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3). Making dynamic geometry software available as well as tracing paper, straightedge, and compass gives students an opportunity to choose appropriate tools strategically (MP5).

Display several students’ inscribed circles for different kinds of triangles for all to see. 

Ask students: “How do we know that an inscribed circle is tangent to all 3 sides of the triangle?” Give students one minute to record an answer to this question.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response by correcting errors, clarifying meaning, and adding details.

• Display this first draft: “The circle touches all the sides.” 
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Display and review these criteria: 
  • Use specific words and phrases: “tangent,” “inscribed circle,” and “incenter.”
  • Use complete sentences.
  • Refer to a labeled diagram.
• Here is an example of a second draft: “In the diagram, the radius is perpendicular to the side . It has to be perpendicular because it shows the distance from the incenter to the side. The reference chart says: ‘A line is tangent to a circle if and only if it is perpendicular to the radius drawn to the point of tangency.’ So is tangent to the inscribed circle. The same argument works for both the other sides.” 
  

  

• Select 1–2 students to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

The three angle bisectors of a triangle meet at a single point, called the triangle’s incenter. This point is the center of the triangle’s inscribed circle. (Theorem)

Students prove that in an equilateral triangle, the incenter and the circumcenter coincide. Monitor for students who use triangle congruence and for those who use transformations.

If students aren’t sure how to start, ask them what needs to be true about the segments and in order for point to be the triangle’s circumcenter. (The circumcenter is the single point equidistant from all the vertices, so if segments and are all congruent, then is the circumcenter.)

Then ask students:

• “Are there any auxiliary lines you can add to the diagram?”
• “Are there any transformations that might be helpful?”
• “Do any triangle congruence theorems apply?”
• “Are there any angle markings you can add to the diagram?”

The purpose of this discussion is for students to consider multiple strategies for proving that the incenter of an equilateral triangle is also the circumcenter. Invite students to share their reasoning. If possible, select a student who used triangle congruence and another who used transformations. Then ask students:

• “Why doesn’t this proof work for a triangle that is not equilateral?”
  • The angle measures in the original triangle wouldn’t be congruent, so the 6 small triangles created by drawing segments representing the distance from to the triangle’s sides wouldn’t all be congruent.
  • If and don’t have the same length, the perpendicular bisector of won’t pass through .
• “What will the inscribed and circumscribed circles look like for this triangle? Use your compass to construct them.” (They will be concentric circles.)
• “How would the perpendicular bisectors of a triangle’s sides relate to the lines already drawn?” (They would overlap the segments showing the distance from point to the sides of the triangle.)