nine-point circle

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Consider a triangle with vertices A, B and C. Let A′, B′ and C′ be the midpoints of the three sides. Let D, E and F be the feet of the perpendiculars from A, B and C, respectively, to the opposite sides. Then AD, BE and CF are concurrent at the orthocentre H. Let P, Q and R be the midpoints of AH, BH and CH. The points A′, B′, C′, D, E, F, P, Q and R lie on a circle called the nine-point circle. The centre of this circle lies on the Euler line of the triangle at the midpoint of OH, where O is the circumcentre. Feuerbach's Theorem states that the nine-point circle touches the incircle and the three excircles of the triangle. This was proved in 1822 by Karl Wilhelm Feuerbach (1800–34).

Subjects: Mathematics.

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