In Euclidean-geometry the centroid, orthocenter and circumcenter of a triangle lie on a line.
In which other geometries does this hold ?
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
5
|
|
|
|
|
5
|
In the paper "On some classical constructions extended to hyperbolic geometry", A. V. Akopyan proves an analogue of Feuerbach's theorem for hyperbolic geometry. Let $M_a,M_b,M_c$ be three points on sides of a triangle so that the corresponding cevians bisect the area of the triangle $ABC$. The intersection of these cevians, G, is the analogue of the centroid. Next if we take the circle passing through $M_a,M_b,M_c$, it will intersect each side a second time, in points $H_a,H_b,H_c$. The intersection of $AH_a,BH_b,CH_c$ is the analogue of the orthocenter, call it $H$. The paper proves that there is a line passing through $G$, $H$, the circumcenter and the center of the circle passing through $M_a,M_b,M_c$. This is pretty much the most natural analogue for the Euler line in hyperbolic geometry. |
||||||||
|
You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
|
0
|
Only in Euclidean geometry. |
||||||||||||||||
|
