# Droz-Farny Line Theorem

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In 1899, Arnold Droz-Farny (1865-1912), a Swiss science and mathematics teacher, published without proof the following theorem:

If two perpendicular straight lines are drawn through the orthocenter of a triangle, they intercept a segment on each of the three sidelines. The midpoints of the three segments are collinear.

A synthetic proof of the theorem along with a bibliography and a short bibliographical note on Droz-Farny has recently appeared in the *Forum Geometricorum*. The theorem is curious, but the proof is absolutely remarkable in its simple elegance.

### Proof (J.-L. Ayme, 2004)

The statement of the theorem is quite obvious in two cases: when the given triangle is right and, for an arbitrary triangle, when one of the given lines coincides with one of the altitudes of the triangle. In the following we shall ignore the two possibilities.

Consider ΔABC. Assume the given lines meet the side AB of the triangle in points Z and Z', the side BC in points X, X', and the side AC in points Y and Y'. Let M_{a}, M_{b}, M_{c} denote the midpoints of segments XX', YY', and ZZ', respectively. The Droz-Farny theorem states that the points M_{a}, M_{b}, M_{c} are collinear.

Let C_{ABC} denote the circumcircle of ΔABC. A similar notation will be used for other triangles, e.g. C_{XX'H} will denote the circumcircle of ΔXX'H, etc.

The reflection K_{a} of the orthocenter H in BC lies on C_{ABC}. Since the two given lines are perpendicular, the circle C_{XX'H} has XX' as its diameter (and M_{a} as its center.) The circle is symmetric in XX', hence in BC. So that the reflection K_{a} of the orthocenter H in BC lies on C_{XX'H} and serves as a common point of the latter with the circumcircle C_{ABC}. Similar considerations apply to points K_{b} and K_{c} and circles C_{YY'H} and C_{ZZ'H}.

The reflections of line XYZ in the sides of ΔABC meet on C_{ABC} in, say, point Q. The crucial step in the proof to observe we now can apply Miquel's Pivot theorem to each of the three triangles: XYQ, YZQ, and XZQ.

Indeed, we may think of points K_{a}, K_{b}, and H as lying on the sidelines XQ, YQ, and, respectively, XY of ΔXYQ. Circle C_{XX'H} passes through X, K_{a}, and H. Circle C_{YY'H} passes through Y, K_{b}, and H. And, finally, circle C_{ABC} passes through K_{a}, K_{a}, and Q. According to the Pivot theorem the three circles have a common point, say, M. Thus, besides sharing point K_{a}, circles C_{XX'H} and C_{ABC} meet at M, while circles C_{YY'H} and C_{ABC} meet in M and K_{b}.

If we started with, say, ΔYZQ, the Pivot theorem would lead to a point N common to circles C_{YY'H}, C_{ZZ'H}, and C_{ABC}. But circles meet only in two points of which one is K_{b}. We are bound to conclude that _{XX'H},C_{YY'H}, C_{ZZ'H} meet in M and H, which means they are coaxal. In particular, their centers M_{a}, M_{b}, and M_{b} are collinear.

**Note:** The collinearity holds when the three points M_{a}, M_{b}, and M_{b} are defined by the same linear combination of the points of intersection: _{a} = tX + (1-t)X',_{b} = tY + (1-t)Y',_{c} = tZ + (1-t)Z',

### References

- J.-L. Ayme,
__A Purely Synthetic Proof of the Droz-Farny Line Theorem__,*Forum Geometricorum*, Volume 4 (2004) 219-224

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