The mysticism of Feuerbach circle (Circulus mysticus Feuerbachi)

Abstract

If we are given a triangle ABC and define its orthocenter D in it (that is, the common point of intersection of the perpendiculars from its vertices to the opposite sides), then ABCD can be taken as a complete quadrangle with pairs of opposite sides AB and CD, BC and DA and CA and BD . Drawing a circle through the bases (a₁, b₁, c₁) of the perpendiculars on the sides of the triangle, we obtain a Feuerbach circle, which, in addition to these three points, will pass through six more midpoints of the just listed sides of the complete quadrilateral, that is, points a₁, b₁, c₁, a'₁, b'₁, c'₁.

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