两点间的中点（或其它比例点）

The task of finding midpoints in multiple dimensional coordinate spaces is analogous to the similar task on the real line. For example, in 1-D, the midpoint of $0$ and $2$ is just $1$; generally, the midpoint of $a$ and $b$ is their average $\frac{a+b}{2}$.

Similarly, in two dimensions, the midpoint is found by finding the average of the two points in each dimension. If point $A$ has x-coordinate $0$ and point $B$ has x-coordinate $2$, then it stands to reason their midpoint $M$ would have x-coordinate $1$. A similar line of reasoning holds for the y-coordinate.

Of course, this extends to even more dimensions.

In general, the midpoint of $A=(x_1,x_2,...,x_n)$ and $B=(y_1,y_2,...,y_n)$ is as follows

$M=(\frac{x_1+y_1}{2},\frac{x_2+y_2}{2},\dots,\frac{x_n+y_n}{2})$.

What's really convenient here is that we didn't need to solve any equations to show it is the midpoint! For any two points $A,B$, we've found a point $M$ lying on the segment $AB$, with $AM=BM$ (verify this last bit with the distance formula to convince yourself). We also know that the "midpoint" is the unique point satisfying these two conditions, so $M$ must be the midpoint! This strategy of "guessing" an answer that works and using uniqueness, implying your guess is the only possible correct answer, is a very powerful one.

Max made a good explanation in finding midpoints of two points in multiple dimensional coordinate spaces. Now, our task is to find the point, $P$ that divides a line in a given ratio, $\lambda$.

Consider a line segment $AB$, with a point $P$ on it. The point $P$ separates the line segment $AB$ into $AP$ and $PB$. We let the ratio of $AP$ and $PB$ be $\lambda$. Then,

$$\lambda=\frac{AP}{PB}$$

Given that the endpoints of the line segment $AB$ are $A(x_1,y_1)$ and $B(x_2,y_2)$. Let the coordinate of point $P$ be $P(x,y)$. Since

$$\lambda=\frac{AP}{PB}$$

Then

$$\lambda=\frac{x-x_1}{x_2-x}$$

Solve for $x$, we get

$$x=\frac{x_1+\lambda x_2}{1+\lambda}$$

Similarly,

$$y=\frac{y_1+\lambda y_2}{1+\lambda}$$

In conclusion, the coordinate of point $P$ that divides the line $AB$ with endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ in a given ratio, $\lambda$ is

$$x=\frac{x_1+\lambda x_2}{1+\lambda},y=\frac{y_1+\lambda y_2}{1+\lambda}$$