Feuerbach's Theorem

1. The Circle which passes through the feet of the Perpendiculars dropped from the Vertices of any Triangle on the sides opposite them passes also through the Midpoints of these sides as well as through the Midpoint of the segments which join the Vertices to the point of intersection of the Perpendicular (a Nine-Point Circle).

2. The Nine-Point Circle of any Triangle is Tangent internally to the Incircle and Tangent externally to the three Excircles.

See also Excircle, Feuerbach Point, Incircle, Midpoint, Nine-Point Circle, Perpendicular, Tangent

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 117-119, 1967.

Dixon, R. Mathographics. New York: Dover, p. 59, 1991.