Feuerbach theorem - full construction
Thm: a triangle ABC is given.
1. The feet of the altitudes, the midpoints of the sides and the midpoints of the segments joining the vertices to the point of intersection of the altitudes (orthocentre) are concyclic. The circle that they determine is called the 9-point circle.
2. The 9-point circle is tangent to the incircle and to the 3 external incircles of ABC.
This applet helps to verify the theorem without advanced commands such as AreConcyclic.