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:
“You treat world history as a mathematician does mathematics, in which nothing but laws and formulas exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.”
—Hermann Hesse (1877–1962)
“The reputation of generosity is to be purchased pretty cheap; it does not depend so much upon a man’s general expense, as it does upon his giving handsomely where it is proper to give at all. A man, for instance, who should give a servant four shillings, would pass for covetous, while he who gave him a crown, would be reckoned generous; so that the difference of those two opposite characters, turns upon one shilling.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)