The coordinates of the vertices of a triangle are $(x_{1}, y_{1})$ , $(x_{2}, y_{2})$ and $(x_{3}, y_{3})$ . The line joining the first two is divided in the ratio $l : k$ , and the line joining this point of division to the opposite angular point is then divided in the ratio $m : k + l$ . Find the coordinates of the latter point of section.

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Solution

Using the section formula, if a point $(x, y)$ divides the line joining the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in the ratio $m : n$ , then
$(x, y) = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})$
The vertices of the triangle are given to be $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ . Let these vertices be $A, B$ and $C$ respectively.
Then the coordinates of the point $P$ that divides $A B$ in $l : k$ will be
$(\frac{l x_{2} + k x_{1}}{l + k}, \frac{l y_{2} + k y_{1}}{l + k})$
The coordinates of point which divides $P C$ in $m : k + l$ will be
$⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{m x_{3} + (k + l) \frac{l x_{2} + k x_{1}}{(l + k)}}{m + k + l}, \frac{m y_{3} + (k + l) \frac{l y_{2} + k y_{1}}{(l + k)}}{m + k + l} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ \Rightarrow (\frac{k x_{1} + l x_{2} + m x_{3}}{m + k + l}, \frac{k y_{1} + l y_{2} + m y_{3}}{m + k + l})$

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