The mysticism of the Feuerbach circle (Circulus mysticus Feuerbachi)

Abstract

If a trigon ABC is given and we define the height point D in it (that is, the common point of intersection of the perpendiculars from its vertices to opposite sides), then ABCD can be taken as complete quadrangles with pairs of opposite sides AB with CD, BC with DA and CA with 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.