Date: Dec 28, 1997 3:26 AM
Author: Michael Keyton
Subject: Re: Orthocentre
The orthocenter is one of the more important points of a triangle. Of the
4 major centers, it is the only one not mentioned in Euclid.
One could write a book about this point, there are so many interesting
aspects of it. I taught a class on euclidean geoemtry for high school
teachers last year, and opened each class with a property of the
orthocenter, and I just scratched the surface.
just a few.
Possibly the most important aspect: take a triangle with vertices at T, R,
and I, and let H be its orthocenter. The orthocenter for any of the
triangles formed from three of these four points is the fourth point. (4
points which satisfy this condition are called an orthocentric set.)
Bisect the exterior angles of a triangle, they intersect at three points,
called the excenters. The incenter of the triangle is the orthocenter of
the triangle formed by the 3 excenters. Thus, the incenter and the 3
excenters form an orthocentric set. (There are many other orthocentric
From this it then follows: let O, C, and E be the feet of the altitudes of
triangle TRI. Triangle OCE is called the orthic triangle of TRI, the
altitudes bisect the angles of the orthic triangle; i.e., the incenter of
the orthic triangle is the orthocenter of the original tirangle.
The orthocenter, the centroid, and the circumcenter are collinear, the
centroid is a trisection point. The line through these 3 points is called
the Euler line. (There are many proofs of this, some surprisingly quite
simple, one using dilations makes the theorem obvious (both aspects).
The circumcircle of the orthic triangle contains the midpoints of the
sides of the triangle, and the midpoints from the othocenter to each of
the vertices of the triangle. (This circle is called the 9 point circle,
though, there are now known over 40 points that reside on it.) (It is also
called the Feuerbach circle. It is tangent to the incircle and the three
excircles.) The center of this circle is the midpoint between the
orthocenter and the circumcenter; it is called the 9 point circle center.)
The distance from the orthocenter to a vertex is half the distance from
the circumcenter to the opposite side of the vertex.
You mentioned reflecting the orthocenter over the sides of the triangle
proces points that live on the circumcircle of the triangle.
Reverse this process, (which is very impressive on Cabri) take any point
on the circumcircle, reflect it over the sides of the triangle. They are
collinear. Take the locus of these points as the point moves about the
circumcircle. These describe three circles which are concurrent at the
orthocenter of the triangle.
This is a few of the interesting aspects of the orthocenter. If you are
interested in others, I can supply you with several excellent references.
On 27 Dec 1997, Hai Trung Ho wrote:
> I only know that the orthocentre, reflected on one of the triangle's
> sides will lie on the circumcircle of the triangle. Are there any
> other properties of the orthocentre???
> Hai Trung Ho.