Miquel's Theorem

Strategies and Hints

  1. It will be hard for students to locate the centers of the three circles.

  2. Students can discover the property in the solution by experimenting with circles of different sizes.

  3. This property allows you to draw the circle that passes through points E, D, and C. The other two circles are found in the same way. By using this property, you can construct the figure as it is described in the problem.

Solution

No matter what triangle is chosen, and no matter what points are selected on the sides of the triangle, the three circles intersect in a common point, 0.

The theorem is stated as follows: If a point is selected on each side of a triangle, then the circles determined by each vertex and the points on the sides that intersect at that vertex pass through a common point.