This short lesson was inspired by one of the problems from the blog Five Triangles Mathematics. The author challenges the reader to construct a circle using only a compass and straight edge, through two points X and Y. The centre of the circle must be a point on the line located between the two given points. If you can’t visualise it, click here to see the diagram and try the problem first and then come back if you are interested to see how you might teach this in your class without losing the essence of the problem solving activity.
Here’s my sequence of tasks. Notice that all three tasks involve geometric constructions in increasing complexity, one building on the previous task.
You can use this as context for reviewing the properties of isosceles triangle after the students have come up with at least two solutions.
This is one of the solution but I suggest you ask students to come up with other ways of constructing the isosceles triangle. The procedure shown involved constructing the perpendicular bisector of CD. F is any point on the perpendicular bisector.
(Of course I hid some part of the construction to make it a little bit of a challenge. Do you think the location of J is unique?)
In terms of time, this is not actually a short lesson because you need to give students more time to solve the problems. You may also want to read How to scaffold problem solving in geometry. The following book is a good resource for tasks that fosters geometric thinking.