```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 the4 major centers, it is the only one not mentioned in Euclid.One could write a book about this point, there are so many interestingaspects of it. I taught a class on euclidean geoemtry for high schoolteachers last year, and opened each class with a property of theorthocenter, 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 thetriangles formed from three of these four points is the fourth point. (4points 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 ofthe triangle formed by the 3 excenters. Thus, the incenter and the 3excenters form an orthocentric set. (There are many other orthocentricsets.) From this it then follows: let O, C, and E be the feet of the altitudes oftriangle TRI. Triangle OCE is called the orthic triangle of TRI, thealtitudes bisect the angles of the orthic triangle; i.e., the incenter ofthe orthic triangle is the orthocenter of the original tirangle.The orthocenter, the centroid, and the circumcenter are collinear, thecentroid is a trisection point. The line through these 3 points is calledthe Euler line. (There are many proofs of this, some surprisingly quitesimple, one using dilations makes the theorem obvious (both aspects).The circumcircle of the orthic triangle contains the midpoints of thesides of the triangle, and the midpoints from the othocenter to each ofthe 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 alsocalled the Feuerbach circle. It is tangent to the incircle and the threeexcircles.) The center of this circle is the midpoint between theorthocenter and the circumcenter; it is called the 9 point circle center.)The distance from the orthocenter to a vertex is half the distance fromthe circumcenter to the opposite side of the vertex.You mentioned reflecting the orthocenter over the sides of the triangleproces points that live on the circumcircle of the triangle.Reverse this process, (which is very impressive on Cabri) take any pointon the circumcircle, reflect it over the sides of the triangle. They arecollinear. Take the locus of these points as the point moves about thecircumcircle. These describe three circles which are concurrent at theorthocenter of the triangle.This is a few of the interesting aspects of the orthocenter. If you areinterested in others, I can supply you with several excellent references.Michael KeytonOn 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.>
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