中线带来的长度比

Consider the diagram, where $D,E,F$ are midpoints of the sides of triangle ABC. The medians concur at the centroid G.

We use a technique called mass points to prove this. Mass points consists of assigning "weights" to certain vertices on a triangle, and making all the weights balance at a point in a triangle.

It makes sense that if a is the mass on point A, b is the mass point B, and C is in the middle, then for everything to balance we must have $a \cdot BC = c \cdot AB$.

Thus, we assign masses of $1,1,1$ to points $A,B,C$.

It's 2 because if there are weights 1 and 1 on B and C, then these weights combine and $1+1=2$.

Thus, $D,E,F$ all have weights of 2. Finally G has a weight of $2+1=3$. This means that the ratios $BG:GE,CG:GF,AG:GD$ are all 2 to 1.

We also have a different proof of this fact:

Yes... so we know $[ACF]=[BCF], [AGF]=[BGF]$. We subtract to get $[AGC]=[BGC]$

Since $[AGC]=2[AEG],[BGC]=2[BGD]$, we have $[AEG]=[BGD]$.

By this logic, all six triangles have equal area.

Now, ratio $[ABC]:[AGB]$ is three to one. Since the bases are equal, this implies the heights have ratio three to one! We can draw the altitude form C to AB. It's not hard to find that if the ratio of the altitudes is 3 to 1, then the ratio $CF$ to $GF$ is three to one, so we are done.

In the above diagram, $E,D,F$ are the midpoints of $AC, BC, AB$ respectively. As you probably know, $AD, BE, CF$ are called the medians of $\Delta ABC$, and $G$, the intersection of the medians, is the centroid.

We'll try to prove these conjectures now.

Because $AF=FB$, and the altitudes are equal, $AGF$ and $GFB$ have the same area. Let's denote that area $x$. Similarly, the area of $CDG=GDB$, which we'll call $y$, and $ECG=AEG$, which we'll denote as $z$ .

It would be very convenient if $x=y=z$, as that would make the area of $ABC=6x$ and $AGB=BGC=AGC=2x$, which would finish our proof. But we can prove this!

Because $AF=FB$, we have the area of $ACF=BCF$. This means that $$\begin{align*} x+2z &= x+2y \\ 2z&=2y \\ z&=y \end{align*}$$ By changing the side of the triangle we use as the base, we get $x=y$ and $x=z$, so $x=y=z$ and we're done!