Math Is Fun Forum

vlnikolic

Member

Registered: 2010-09-02

Posts: 5

simplex splitting using center of gravity

asd

Last edited by vlnikolic (2010-09-27 20:13:47)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: simplex splitting using center of gravity

Hi vlnikolic,

Triangle case.

I'm assuming a diagram like image two below;  ie. G lies at the intersection of the medians.

And you want to prove that image one cannot occur.

Consider the diagram in image three.

ABC is the triangle,   D, E and F are the midpoints of BC, AC and AB.

Join D to F and let it intersect BE at G.

BDF is similar to BCA (ie. same angles) and half the size, so BG = ½ BE.

Angle FGD = 180

For angle FGD to be reflex, G must be nearer to B than half way along BE.

But the centre of gravity, G, is only one third of the way up the median so this will never happen.

Bob

Last edited by Bob (2010-09-03 03:56:17)

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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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: simplex splitting using center of gravity

hi vlnikolic

I cannot be sure that 3D works this way.  My problem with that case is I cannot get a clear picture in my mind of the solids you are making.  And also, what do you mean by 'convex' for a 3D solid.  Have you ever used google sketchup.  It's a free download and then you could make your solids and rotate them.  And send me a copy of the file.  Let me know if you want to try this.

Bob

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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

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: simplex splitting using center of gravity

hello vlnikolic

I'm going to try to make a model using Sketchup.  Then I'll need a big think so I may be gone some time.

Bob

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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