Feuerbach

Feuerbach's theorem

The Euler-circle of the triangle ABC is tangent to the tritangent circles of the triangle.

The proof suggested by the figure below, proceeds through inversion w.r. to the circle with diameter KL (projections of the centers I, J on BC). This inversion interchanges the Euler circle with the line B*C*, which is the symmetric of BC w.r. to IJ. Later is proved by showing that
A'C' * A'C'' = A'B' * A'B'' = A'K^2.

Since B*C* is tangent to the circles with centers I and J, and these circles remain invariant w.r. to the above mentioned inversion, the image of B*C*, which is Euler's circle, will be also tangent to the two circles.

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