### Cantor's Theorem: What Is It?

A Mathematical Droodle

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Copyright © 1996-2017 Alexander Bogomolny

The applet suggests what is known as Cantor's theorem (M. B. Cantor, 1829-1920):

Perpendiculars from the midpoints of a triangle to the opposite sides of its tangential triangle are concurrent. Furthermore, they meet at the center of the nine-point circle.

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(*Tangential triangle* is formed by the tangents to the circumcircle at the vertices of the given triangle. The lines in the theorem are known as *Cantor's lines* and their common point is sometimes referred to as the *Cantor's point*.)

In complex variables (or affine geometry, where we can add points) the proof is truly simple. Let the origin is set at the circumcenter O of ΔABC. Let z_{A}, z_{B} and z_{C} be the complex numbers corresponding to its vertices. The midpoints of the sides are given as (side centroids or barycenters) _{A} = (z_{B} + z_{C})/2,_{A} + z_{B} + z_{C})/2._{A}N = N - m_{A} = z_{A}/2_{A} drawn to the vertex A. It is thus perpendicular to the tangent to the circumcircle at A. We see that Cantor's perpendicular from m_{A} passes through N, the nine-point center. The same holds for the other two lines.

### Remark

Cantor actually proved a more general statement. For any inscribed n-gon we may consider n lines, one per vertex. Pick a vertex p and the centroid g of the remaining vertices. From g draw the perpendicular to the tangent at p. *Cantor's theorem* asserts that all n lines always concur. The point of concurrence is known as *Cantor's point* of the polygon.

### References

- R. Honsberger,
*Mathematical Chestnuts from Around the World*, MAA, 2001, pp 241-244

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