Feuerbach's Theorem: What Is It About?
A Mathematical Droodle

Explanation

Copyright © 1996-2009 Alexander Bogomolny

This is a famous result discovered by Karl Wilhelm Feuerbach (1800-1834): the 9-point circle of a triangle touches its incircle and the three excircles. Feuerbach proved the theorem by computing algebraically the distances between the centers of the circles. Other proofs depend on the inversion in a circle. The point of tangency of the incircle and the 9-point circle is known as the Feuerbach point. The 9-point circle is known to house a few dozen other intereseting points. The points of tangency in Feuerbach's theorem are involved in two cases of incidence.

Nine Point Circle

• Feuerbach's Theorem
• Feuerbach's Theorem: a Proof
• Four 9-Point Circles in a Quadrilateral
• Four Triangles, One Circle
• Hart Circle
• Incidence in Feuerbach's Theorem
• Six Point Circle
• Nine Point Circle
• 6 to 9 Point Circle
  • Six Concyclic Points
• Six Concyclic Points II
• Bevan's Point and Theorem

Copyright © 1996-2009 Alexander Bogomolny