$A(x_{1}, y_{1})$, $B(x_{2}, y_{2})$ and $C(x_{3}, y_{3})$ are three points, and the condition for the collinearity of them is

$\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}} \,=\, \dfrac{y_{3}-y_{2}}{x_{3}-x_{2}}$

$A(x_{1}, y_{1})$, $B(x_{2}, y_{2})$ and $C(x_{3}, y_{3})$ are three points on a straight line in two dimensional Cartesian coordinate system. The three points are known as collinear points geometrically. Let’s try to derive a condition for the collinearity of three points mathematically.

Draw parallel lines to horizontal $x$-axis from points $A$ and $B$. Similarly, draw lines from points $B$ and $C$ but they should be perpendicular to $x$-axis. The parallel and perpendicular lines get intersected at points $D$ and $E$.

It forms two right triangles, known as $\Delta DAB$ and $\Delta EBC$ in two dimensional space.

Geometrically, $\overline{AD} \,\|\, \overline{BE}$ and the two line segments $\overline{AB}$ and $\overline{BC}$ are part of the straight line. Therefore, $\angle DAB$ and $\angle EBC$ are congruent. If $\angle DAB = \theta$, then $\angle EBC = \theta$.

Find the slope of the straight line $\overleftrightarrow{AB}$ from right triangle $DAB$.

$\tan{\theta} \,=\, \dfrac{BD}{AD}$

$\implies \tan{\theta} \,=\, \dfrac{OB-OD}{OD-OA}$

$\,\,\, \therefore \,\,\,\,\,\, \tan{\theta} \,=\, \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Find the slope of the straight line $\overleftrightarrow{BC}$ from right triangle $EBC$.

$\tan{\theta} \,=\, \dfrac{EC}{BE}$

$\implies \tan{\theta} \,=\, \dfrac{OC-OE}{OE-OB}$

$\,\,\, \therefore \,\,\,\,\,\, \tan{\theta} \,=\, \dfrac{y_{3}-y_{2}}{x_{3}-x_{2}}$

We know that $\overline{AB}$ and $\overline{BC}$ are part of the straight line $\overleftrightarrow{AC}$. So, the slopes of straight lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{BC}$ are equal.

$\,\,\, \therefore \,\,\,\,\,\, \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}} \,=\, \dfrac{y_{3}-y_{2}}{x_{3}-x_{2}}$

The mathematical equation in algebraic form represents a condition for the collinearity of three points and it is useful to us to verify the collinearity of three or more points mathematically.

Latest Math Topics

Jul 20, 2023

Jun 26, 2023

Jun 23, 2023

Latest Math Problems

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved