Droz-Farny Circles

Arnold Droz-Farny (1865-1912), a discoverer of a remarkable line now bearing his name, has also established the congruence of two related circles:

Droz-Farny 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

which we apply to the median P_aM in triangle P_aPQ.

Let Y be one of the intersections of BC with the circle through Q centered at P_a, so that

In the triangle P_aPQ, the Pythagorean theorem gives:

In the last expression, P_aM 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

1. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, pp. 69-72 (2004) 219-224

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