DrozFarny Circles
Arnold DrozFarny (18651912), a discoverer of a remarkable line now bearing his name, has also established the congruence of two related circles:
What if applet does not run? 
Suppose that P and Q are any pair of isogonal conjugates in ΔABC and that P_{a}, P_{b}, and P_{c} are the feet of perpendiculars from P to the sides of the triangle. Assume also that the circles with centers P_{a}, P_{b}, and P_{c} are drawn to pass through Q. Then the three pairs of points these circles determine on the side line of ΔABC will always lie on the circle with center P. A circle with center Q is obtained in a similar manner. The two circles always have the same radius. 
DrozFarny has proved the theorem for a special pair of isogonal conjugates: O (circumcenter) and H (orthocenter) of a triangle, but it is as easily proven in a more general case.
The proof is based on a metric property of medians in a triangle, that is convenient to write as
b^{2} + c^{2} = 2m_{a}^{2} + a^{2}/2, 
which we apply to the median P_{a}M in triangle P_{a}PQ.
Let Y be one of the intersections of BC with the circle through Q centered at P_{a}, so that
P_{a}Y = P_{a}Q. 
In the triangle P_{a}PQ, the Pythagorean theorem gives:
(1) 
.

In the last expression, P_{a}M is the radius of the pedal circle of P with respect to ΔABC and is, thus, independent of being associated with the side BC. PY, therefore, would be exactly the same if, instead of a circle with center at P_{a}, we chose circles with centers on sides AB or AC. We conclude that there is a circle centered at P that passes through the six points cut on the sides of the triangle by the circles through Q centered at the feet of perpendiculars from P. But now the same holds with regard to point Q: there is a circle through the six points cut by the circles through P centered at the feet of perpendiculars from Q to the sides. Furthermore, since the pedal circles of a pair of isogonal conjugates coincide, (1) produces the same result for the two circles.
References
 R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, pp. 6972 (2004) 219224
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
65255479 