### Nagel's Theorem: What is it?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander Bogomolny

The applet attempts to suggest Nagel's theorem:

Let AD and BE be two altitudes in ΔABC and O its circumcenter. Then DE and CO are perpendicular. |

Since DE is a side of the orthic triangle, the statement of Nagel's theorem is equivalent to the assertion that the sides of the orthic triangle are perpendicular to the radius-vectors of the circumcircle drawn to the corresponding vertices of the reference triangle.

What if applet does not run? |

In any triangle, circumcenter and orthocenter are isogonal conjugate. In other words, the altitudes and suitable circum-radius-vectors are reflections of each other in the corresponding angle bisectors. Also, the sides of the orthic triangle are antiparallels relative to the opposite sides of ΔABC, which also means that the directions of the sides of the orthic and reference triangles are reflections in the suitable angle bisectors. Since, the altitudes are perpendicular to the sides of the reference triangle, it follows that the circum-radius-vectors are perpendicular to the sides of the orthic triangle.

(There is another elementary proof.)

### References

- R. A. Johnson,
*Advanced Euclidean Geometry (Modern Geometry)*, Dover, 2007, p. 172

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