Four Triangles, One Circle: What Is It About?
A Mathematical Droodle

Explanation

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Copyright © 1996-2012 Alexander Bogomolny

Let H be the orthocenter of ΔABC. Then A is the orthocenter of ΔBCH, and similarly for B and C. This observation leads to a curious fact that all four triangles share the 9-point circle. In ΔABC the 9-point circle passes through the midpoints of the sides, the feet of the altitudes and the Euler points, i.e., the points midway from the orthocenter to the vertices.

Now, by Feuerbach's theorem, the 9-point circle is tangent to the incircle of a triangle and its excircles. It follows, that in a configuration of four points A, B, C, H and four triangles ABC, ABH, BCH, CAH, the same circle (the 9-point circle of all four triangles) is tangent, in general, to sixteen circles:

Nine Point Circle

• Nine Point Circle: an Elementary Proof
• 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

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Copyright © 1996-2012 Alexander Bogomolny