Students will learn about Thales's theorem and its converse. They must already know the isosceles triangle theorem. Our first goal for students, is to have them make the conjecture. Have students begin by opening this file in Geogebra:

Tell them they are allowed to move point \(C\) anywhere along the arc \(AB.\) Ask them what they notice. Alternatively, you could give them a sheet of various triangles in semicircles and a protractor. The conjecture students make may be wrong, but either way, have them attempt to prove it. If they fail to make a conjecture, nudge them along by telling them to concentrate on angle \(ACB.\) After that, challenge your students to prove Thales's theorem. Students are most likely to discover a proof similar to this one. After students have found a proof of Thales's theorem, with or without your help, have them open this Geogebra file:

This is nice because it lets students verify the claim for any position of \(C\) they like. It bolsters their confidence that the theorem is true. Doing this isn't strictly necessary, but I think it makes the experience more enjoyable. Next, show your students this proof for the converse of Thales's theorem. Finally, ask students how they could determine the diameter of a circle, given only an additional piece of paper, a pencil, and a ruler. The answer is to place the piece of paper on the circle, such that the corner of the paper lies on the boundary of the circle. Then mark the two points where the edges of the paper cross the circle, using the pencil. Then measure the distance between those points using the ruler (source.)

Could students discover the proof for the converse of Thales's theorem too, or would that be too difficult for the vast majority of students?

Conclude by giving your students these challenges:

- Sum Equals Product by NRICH
- 2017 Math Kangaroo Level 5-6 Problem #25 by STEM4all
- 2019 Math Kangaroo Levels 11-12 Problem #16 by STEM4all
- As Easy as 1,2,3 by NRICH
- Tangram Money by Pierce School: Problem / Solution