# Fate of the Euler Line and the Nine-Point Circle on the Sphere

Fate of the Euler Line and the Nine-Point Circle on the Sphere

In Euclidean geometry, the circumcenter, the centroid and the orthocenter of a triangle always lie on a line called the Euler line. Correspondingly, the midpoints of the sides, the feet of the altitudes and the midpoints between the vertices and the orthocenter lie on a circle called the nine-point or Euler circle. The nine-point circle is tangent to the incircle and to the excircles of the triangle, and its center also lies on the Euler line. These results do not apply unchanged to spherical geometry.

In spherical geometry, define the pseudo-medians to be the line segments dividing the triangle into two equal-area triangles and the pseudo-centroid to be the concurrence point of the pseudo-medians. Define the pseudo-altitudes to be the line segments dividing the triangle into two triangles with adjacent base angles equal to the sum of the other two angles diminished by half of the spherical excess of the whole triangle and the pseudo-orthocenter to be the concurrence point of the pseudo-altitudes.

Then the circumcenter, the pseudo-centroid and the pseudo-orthocenter always lie on a line called the Akopyan line.

Correspondingly, the feet of the pseudo-medians and of the pseudo-altitudes lie on a circle called the Akopyan circle. This circle is tangent to the incircle and to the excircles of the spherical triangle, and its center lies on the Akopyan line.

In the limit of large radii of curvature, the Akopyan line and circle approach the Euler line and the nine-point circle, respectively.