Euler

Euler Circle and Line

Properties of the [Euler circle] (or nine-point-circle) of the triangle ABC:
1) Passes through the feet D, E, F of the altitudes of the triangle.
2) Passes through the midpoints G, H, I of the sides of the triangle.
3) Passes through the midpoints N, P, Q of the segments on the altitudes, between their common intersection L (orthocenter) and the vertices.
4) The center of the Euler circle is the middle of the segment OL, joining the center O of the circumcircle to the intersection-point L of the heights. This is the [Euler-Line] of ABC. OL contains also the centroid M of the triangle (intersection point of medians), which divides OL in ratio 2:1.
5) The Euler circle and the circumcircle of the triangle are homothetical, with respect to the orthocenter L, with homothety-modulus 1/2. Thus, the radius of the Euler circle is half the radius of the circumcircle.
6) The Euler circle is tangent to all four tritangent circles of the triangle ABC. (Tritangent circles or inscribed circles are the circles which are tangent to all the sides (or their extensions) of the triangle. There are four such circles for each triangle.)
The last property is Feuerbachs-theorem, the corresponding figure is in the file Feuerbach.html .

- Work with the altitude AD to side BC to find a property independent of this particular altitude.
- Take B* diametral of B. Triangles BCB* and BAB* are right-angled at C and A respectively.
- => AD, B*C both orthogonal to BC, CL, B*A both ortthogonal to AB => ALCB* is a parallelogram.
- OG is parallel and half of B*C = AL.
- => NG is parallel and equal to the radius OA of the circumcircle and triangles KNL, KGO are equal.
- K is the common middle of GN and OL. In the right angled GDN: KG = KD = KN.
- Thus K is equidistand from N, G and D the common distance being r/2, half the circumradius r.
- Apply the same reasoning to the other altitudes to show that circle (K,r/2) passes through the 9 points.
- Notice that triangles ALM and GOM are similar in ratio 2:1, hence the location of M on LO.
- Taking the diametral A* of A build triangle ALA* having G as middle of its side LA*.
- The Euler Circle or nine-point-circle is a special case of a nine-point-conic, passing through the six side-middles of a complete quadrilateral. The special case discussed here results when the four vertices of the complete quadrilateral make an orthocentric tetrade i.e. each point of the four is the orthocenter of the triangle of the other three (see the file NinePointsConic.html for the generalization).

Euler Circle and Line

See Also