More Vector Formulas in A General Tetrahedron
If OABC forms a general tetrahedron with a vertex O as the origin and vectors a, b and c represent the positions of the vertices A, B, and C with respect to O, then the radius of the insphere is given by:
and the radius of the circumsphere is given by:
which gives the radius of the twelve-point sphere:
where:
In the formulas throughout this section, the scalar a2 represents the inner vector product a·a; similarly b2 and c2.
The vector positions of various centers are as follows:
The centroid
The incenter
The circumcenter
The Monge point
The Euler line relationships are:
where T is twelve-point center.
Also:
and:
Read more about this topic: Tetrahedron
Famous quotes containing the words formulas and/or general:
“That’s the great danger of sectarian opinions, they always accept the formulas of past events as useful for the measurement of future events and they never are, if you have high standards of accuracy.”
—John Dos Passos (1896–1970)
“‘A thing is called by a certain name because it instantiates a certain universal’ is obviously circular when particularized, but it looks imposing when left in this general form. And it looks imposing in this general form largely because of the inveterate philosophical habit of treating the shadows cast by words and sentences as if they were separately identifiable. Universals, like facts and propositions, are such shadows.”
—David Pears (b. 1921)