Hello dmw

First a quick rant on my part. If you're going to get proficient in mathematics, you *must *learn to say what you mean, and mean what you say. What you meant to say is:$\displaystyle x_3 = \frac{rx_1+sx_2}{r+s},\;y_3 = \frac{ry_1+sy_2}{r+s}$

not:$\displaystyle x_3 = rx_1+\frac{sx_2}{r}+s,\;y_3 = ry_1+\frac{sy_2}{r}+s$

which is what you (and sa-ri-ga-ma) wrote.

Even if you don't know how to use LaTeX (and there's lots of information on this web-site) there's no excuse for leaving out brackets. What you mean is:x3 = (rx1 + sx2)/(r + s), y3 = (ry1 + sy2)/(r + s)

Next, which way round do the $\displaystyle r$ and $\displaystyle s$ go? The answer is that they 'flip' over.

So that if the point $\displaystyle (x_3, y_3)$ divides the line joining $\displaystyle (x_1,y_1)$ to $\displaystyle (x_2,y_2)$ in the ratio $\displaystyle r:s$, then the $\displaystyle s$ and $\displaystyle r$ 'flip' over in the formula, with the $\displaystyle s$ going with the $\displaystyle (x_1,y_1)$ and the $\displaystyle r$ with $\displaystyle (x_2,y_2)$.

Therefore, in the formula above:$\displaystyle x_3 = \frac{rx_1+sx_2}{r+s},\;y_3 = \frac{ry_1+sy_2}{r+s}$

where $\displaystyle r$ goes with $\displaystyle x_1$ and $\displaystyle y_1$ and $\displaystyle s$ with $\displaystyle x_2$ and $\displaystyle y_2$, the point $\displaystyle (x_3, y_3)$ divides the line joining $\displaystyle (x_1,y_1)$ to $\displaystyle (x_2,y_2)$ in the ratio $\displaystyle s:r$ - **not **$\displaystyle r:s$.

It's easy to see why if you think of $\displaystyle r$ being large compared to $\displaystyle s$.

For instance if $\displaystyle r:s = 10:1$, and $\displaystyle P_3$ divides $\displaystyle P_1P_2$ in the ratio $\displaystyle r:s=10:1$, then $\displaystyle P_3$ is much closer to $\displaystyle P_2$ than it is to $\displaystyle P_1$. So its coordinates will be much nearer $\displaystyle P_2$'s than $\displaystyle P_1$'s. So we shall need more $\displaystyle P_2$ and less $\displaystyle P_1$. Therefore the bigger number ($\displaystyle r$) will go with $\displaystyle P_2$ and the smaller ($\displaystyle s$) with $\displaystyle P_1$. They 'flip' over.

Do you get it now?

Grandad