The side lines of a triangle in Feuerbach's theorem can be inverted into three concurrent circles. The circles define an arcual (curvilinear) triangles. Since inversion preserves angles, the tangency is also inherited by the curvilinear case. The incircle in Feuerbach's theorem remains tangent to the three circles. The same holds for the excircles and then also for the 9-point circle. Therefore the following is a generalization of Feuerbach's theorem.
Let there be three mutually intersecting circles that form eight curvilinear triangles (one of which is infinite.) If the circles are c1, c2 and c3, we may consider the points of intersection Ak, A'k of circles ci and cj, where i, j, k are distinct indices that satisfy
|i + j + k = 6,|
so that, for example, circles c1 and c3 intersect in points A2 and A'2. The triangles A'1A2A3, A1A'2A3, A1A2A'3 are said to be associated with triangle A1A2A3. The associated triangles all share one circular arc with the base triangle. However, each of the 8 triangles formed by the three circles could be picked as the base triangle.
For any selection of the base triangle there is a circle tangent to the circles inscribed into the base and its associated triangles. The circle is called the Hart circle. The statement itself is known as Hart's theorem. Clearly the Hart circle plays the role of the 9-point circle in the curvilinear case.
|What if applet does not run?|
In the applet, the three given circles are drawn gray, the incircle of the base triangle green. (Each of these circle is draggable as are its center and a point on the boundary.) The selection of the base triangle is controlled by the drop down box at the bottom of the applet. You can display the incircles of the associated triangles (red), the Hart circle (blue) and the points of tangency where the circles touch each other.
- J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971
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 II
- Bevan's Point and Theorem
- Another Property of the 9-Point Circle
- Concurrence of Ten Nine-Point Circles
- Garcia-Feuerbach Collinearity
- Nine Point Center in Square
Copyright © 1996-2018 Alexander Bogomolny