The following property of a regular triangle, first given by Steiner and then proved by Droz-Farny (1901), is related to these circles. Draw a circle with center at the orthocenter H which cuts the lines M2M3, M3M1, and M1M2 (where M1, M2 and M3 are the midpoints of their respective sides) at P1,Q1; P2, Q2; and P3, Q3 respectively, then the line segments AQ3, AP3, BP1, BQ1,CQ2, AND CP2 are all equal.
Conversely, if equal circles are drawn about the vertices of a triangle (dashed circles in the above figure), they cut the lines joining the midpoints of the corresponding sides in six points (P1, Q1, P2, Q2, P3, Q3) which lie on a circle whose center is the orthocenter.
