Sam Hardin

2022-07-04

I've gotten to the point where I have the following equation:
$2{x}_{2}{x}_{m}-2{x}_{m}{x}_{1}+2{y}_{2}{y}_{m}-2{y}_{m}{y}_{1}={x}_{2}^{2}-{x}_{1}^{2}+{y}_{2}^{2}-{y}_{1}^{2}$
Would it be mathematically correct to split this into the following two equations
$\left\{\begin{array}{l}2{x}_{2}{x}_{m}-2{x}_{m}{x}_{1}={x}_{2}^{2}-{x}_{1}^{2}\\ 2{y}_{2}{y}_{2}-2{y}_{m}{y}_{1}={y}_{2}^{2}-{y}_{1}^{2}\end{array}$
and treat them as a system of equations? If so, then how would I go about doing this?

Mateo Carson

Expert

Step 1
You can not split this equation. That would be the same as saying $a+b=c+d$ can be split into $a=c$ and $b=d$ .
If the points are ${P}_{1},{P}_{2},$ and ${P}_{m}$ , for ${P}_{m}$ to be the midpoint between ${P}_{1}$ and ${P}_{2}$ , then one way of writing this is that ${P}_{m}$ must be on the line between ${P}_{1}$ and ${P}_{2}$ (i.e., ${P}_{m}=t{P}_{1}+\left(1-t\right){P}_{2}$ for $0\le t\le 1$ ) and ${P}_{m}$ must be at the halfway position (i.e., $t=\frac{1}{2}$ ).
Another way is that $|{P}_{1}-{P}_{m}|=|{P}_{2}-{P}_{m}|$ and $|{P}_{1}-{P}_{m}|=|{P}_{2}-{P}_{1}|/2$ . This way gives you two equations for the two unknowns of the x and y components of ${P}_{m}$ .

Do you have a similar question?