Triangle and Circle Theorems in the Euclidean Plane

prepared by:

Joseph Malkevitch
Department of Mathematics and Computing
York College (CUNY)
Jamaica, New York 11451

email: joeyc@cunyvm.cuny.edu
web page: www.york.cuny.edu/~malk

Euler Line

Leonhard Euler (1707-1783) proved the following amazing theorem:

In an (Euclidean) triangle, the orthocenter E, the centroid F, and the circumcenter H are collinear and the length of segment EF is twice the length of segment FH. Furthermore, the length of segment EH is 3 times the length of segment FH.

Recall that:

The orthocenter of a triangle is the point where the three altitudes meet.

The centroid of a triangle is the point where the three medians of the triangle meet.

The circumcenter of the triangle is the point where the three perpendicular bisectors of the sides meet.

A related theorem is due to Jean-Victor Poncelet (1788-1867) and Charles J. Brianchon (1785-1864). This theorem is also sometime attributed to Karl Wilhelm Feurbach (1800-1834).

Given any (Euclidean) triangle the circle which passes through the three points which are the feet of the perpendiculars to the sides of the triangle also passes through the midpoints of the sides of the triangles, as well as the points which bisect the segments joining the three vertices to the orthocenter of the triangle!

This circle is generally known as the 9-point circle.