Proving Aa+ Bb+ Cc = 0 in a Plane Triangle

[mill]

Homework Statement

Let A, B, C be the vertices of a triangle in the plane and let a, b, c be respectively, the midpoints of the opposite sides. Show that Aa+ Bb+ Cc = 0 (all of them have vector signs on the left).

Homework Equations

definition of plane

The Attempt at a Solution

Drew the picture which would look like a triangle within a triangle. The form Aa... looks most like the component equation but I don't understand how a, b, c equates to x, y, z.

 

[LCKurtz]

You say a,b,c are midpoints of sides then you say they are vectors?? If everything is a vector what does Aa mean? And what are x,y,z? You need to state your problem more carefully.

 

[mill]

That is exactly how the problem appears as written by the professor. A, B, and C form a triangle with a, b, c as midpoints. I meant that in the equation for example Aa has a vector sign hovering over it, but I don't know how to type in that symbol.

 

[LCKurtz]

OK, so you mean $\vec{Aa}+\vec{Bb} +\vec {Cc} = \vec 0$. Here's a hint:$$
\vec{Aa} = \vec{AB} +\frac 1 2 \vec{BC}$$and similarly for the other two.

 

[mill]

Got it thanks.