With three distinct and non-aligned points, we form a plane, so that a straight line is formed with them, they must be aligned.
Consider the points A(1,2), B(3,0), C(4,-1). Placing them on a Cartesian plane we can see that the union will form a straight line, that is, they are aligned.
Joining the three distinct points on a Cartesian plane is an option to check their alignment, but this does not always present a safe answer, as one of the three points may be millimeters off the line formed, which leaves the three points not aligned.
For this reason, when checking whether the three points are aligned, the following condition must be followed:
Points A, B and C belong to the line formed above and point B is common to segments AB and BC, in this case we can apply the following property: Two parallel lines that have a common point are coincident.
Joining this property with the calculation of the coefficients, we will conclude that the points A, B and C will be parallel if the coefficients of the two segments mAB and mBC are equal.
mAB = 0 – 2 = – 2 = – 1
3 – 1 2
MBC = – 1 – 0 = –1 = – 1
4 – 3 1
how badAB = mBC we can say that the three (A, B and C) points are aligned.
Analyzing this example we arrive at the following three-point alignment condition:
Given three distinct points A (xA, yB), B (xB, yB) and C (xC, yC), they will be aligned if, only if the coefficients mAB and mBC are equal.
by Danielle de Miranda
Graduated in Mathematics
Brazil School Team
Analytical Geometry - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/condicao-alinhamento-tres-pontos.htm
