Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, and all orthocentric tetrahedra can be formed in this way.
22.
This common nine-point center lies at the midpoint of the connector that joins any orthocentric point to the circumcenter of the triangle formed from the other three orthocentric points.
23.
This common nine-point center lies at the midpoint of the connector that joins any orthocentric point to the circumcenter of the triangle formed from the other three orthocentric points.
24.
A tetrahedron " ABCD " is orthocentric if and only if the sum of the squares of opposite edges is the same for the three pairs of opposite edges:
25.
Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.
26.
Because the nine-point circle is common to all four possible triangles in an orthocentric system it is tangent to 16 circles comprising the incircles and excircles of the four possible triangles.
27.
The well documented rectangular hyperbolas that pass through four orthocentric points are the Feuerbach, JeY�bek and Kiepert circumhyperbolas of the reference triangle ABC in a normalized system with H as the orthocenter.
28.
The joining of these three orthogonal points into a triangle generates an orthic triangle that is common to all the four possible triangles formed from the four orthocentric points taken three at a time.
29.
Where O 1, O 2, O 3 and O 4 are the circumcenters of the four possible triangles formed from the orthocentric points A 1, A 2, A 3 and A 4.
30.
When " P " is chosen as the circumcenter O, then ? = & minus; 1 and the generated orthocentric system is Johnson triangle of the original reference triangle " ABC ".
