Miquel points

If we mark any point on each side of a triangle, and through draw a circle through the each vertex and the points on the adjacent sides then the circles will be concurrent at a point called the Miquel point. One special Miquel point is the point used to form the pedal triangle.

To observe a GSP sketch that illustrates the position of the Miquel point for various positions of the points on the sides of the triangle click here.

There are a number of interesting observations to be made regarding the Miquel point:

1. The lines from the Miquel points to the marked points on the sides of the triangle make equal angles with the respective sides. Click here to observe this property on a GSP sketch.

2.. The centres of the Miquel circles are the vertices of a triangle similar to the given triangle click here to observe the similar triangle in a GSP sketch.