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thedarktiger

Member

Registered: 2014-01-10

Posts: 91

tetrahedrons :P

In tetrahedron ABCD, \angle ADB = \angle ADC = \angle BDC = 90^\circ. Let a = AD, b = BD, and c = CD.

(a) Find the circumradius of tetrahedron ABCD in terms of a, b, and c. (The circumradius of a tetrahedron is the radius of the sphere that passes through all four vertices, and the circumcenter is the center of this sphere.)

(b) Let O be the circumcenter of tetrahedron ABCD. Prove that \overline{OD} passes through the centroid of triangle ABC.

Oh well. Thanks!

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: tetrahedrons :P

hi thedarktiger,

Hmmm.  Looks like the rest of that lesson on vectors. 

As ADB = ADC = BDC it would be sensible to make D the origin, DA, DB and DC the axes, and a, b and c vectors in the direction of the axes.

[I've put those vectors in bold, but it will be a pain to have to keep doing that so please just remember those are vectors.]

step 1.  To find G, the centroid of ABC.

vector BA = BD + DA = -b + a.  Let E be the midpoint of BA, then BE = ½(-b + a)

Therefore DE = DB + BE = b + ½(-b + a) = ½(a + b)

Similarly, if F is the midpoint of BC, then DF = ½(b + c)

Now to find the vector equation of CE.

To get to any point on CE you have to first go to C, and then a 'certain amount' in the direction of CE.

CE = CD + DE = -c + ½(a + b).  So if I use Greek letter lambda for the 'certain amount'

and in the same way the equation for AF is

These lines cross at G, so the two r values must be equal there, and as a, b and c are in mutually perpendicular directions that means the 'a' components, the 'b' components and the 'c' components must individually be equal.  ie.

The second implies that mu = lambda and the first that both are equal to 2/3

Check the 'a' components:

Thus DG = 1/3(a + b + c)

Let O be a point on DG.  The direction of DG is (a + b + c) so let's say

Now I'd like OA = OD and I happen to notice that if I set 'nu' to 1/2 then

and

These vectors all have the same magnitude

so O is the centre of the sphere and it lies on DG.  phew

Bob

Last edited by Bob (2014-02-24 02:26:10)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: tetrahedrons :P

Thank you so much! I think I got it.

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Good. You can read.