Comments on BARICENTRIC RELATIONS between THREE COLLINEAR POINTS // ҮС КОЛЛИНЕАРНАЙ ТОЧКА икки ардыгар БАРИЦЕНТРИЧЕСКАЙ СЫҺЫАННАР тустарынан комментарийдар

 We will further specify the relationships between the barycentric coordinates of three points on a line, which appear in the Post here. The notations are slightly modified.

          

          Let  $X,\;Y,\;Z$  be three collinear points. The following relations hold :

$\frac{\overline{XY}}{\overline{XZ}}=r\;\tag{1.1}$

$\overrightarrow{OY}=(1-r)\overrightarrow{OX}+r\overrightarrow{OZ}\;\tag{1.2}$

$Y=(1-r)X+rZ\;\tag{1.3}$

and the writing of (1.2)  $\overrightarrow{OY}=r\overrightarrow{OZ}+(1-r)\overrightarrow{OX}$  also says that

$\frac{\overline{ZY}}{\overline{ZX}}=1-r\;\tag{1.4}$

$\frac{\overline{XZ}}{\overline{XY}}=\frac{1}{r}\;\tag{2.1}$

$\overrightarrow{OZ}=\frac{1}{r}\overrightarrow{OY}+(1-\frac{1}{r})\overrightarrow{OZ}\;\tag{2.2}$

$Z=\frac{1}{r}Y+(1-\frac{1}{r})X\;\tag{2.3}$

$\frac{\overline{YZ}}{\overline{YX}}=1-\frac{1}{r}\;\tag{2.4}$

$\frac{\overline{YX}}{\overline{YZ}}=-\frac{r}{1-r}\;\tag{3.1}$

$\overrightarrow{OX}=\frac{1}{1-r}\overrightarrow{OY}-\frac{r}{1-r}\overrightarrow{OZ}\;\tag{3.2}$

$X=\frac{1}{1-r}Y-\frac{r}{1-r}Z\;\tag{3.3}$

$\frac{\overline{ZX}}{\overline{ZY}}=\frac{1}{1-r}\;\tag{3.4}$