Feuerbach's Theorem

There are two theorems commonly known as Feuerbach's theorem. The first states that circle which passes through the feet of the perpendiculars dropped from the polygon 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 polygon vertices to the point of intersection of the perpendicular. Such a circle is called a nine-point circle.


The proposition most frequently called Feuerbach's theorem states that the nine-point circle of any triangle is tangent internally to the incircle and tangent externally to the three excircles. This theorem was first published by Feuerbach (1822). Many proofs have been given (Elder 1960), with the simplest being the one presented by M'Clelland (1891, p. 225) and Lachlan (1893, p. 74).

See also

Excircles, Feuerbach Point, Feuerbach Triangle, Hart Circle, Incircle, Midpoint, Nine-Point Circle, Perpendicular, Tangent

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Feuerbach's Theorem

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Weisstein, Eric W. "Feuerbach's Theorem." From MathWorld--A Wolfram Web Resource.

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