Three unaligned points on a Cartesian plane form a triangle of vertices A(x)THEyTHE), B(xByB) and C(xÇyÇ). Your area can be calculated as follows:
A = 1/2. |D|, that is, |D| / 2, considering D = 
.
For the area of the triangle to exist, this determinant must be different from zero. If the three points, which were the vertices of the triangle, are equal to zero, they can only be aligned.
Therefore, we can conclude that three distinct points A(xTHEyTHE), B(xByB) and C(xÇyÇ) will be aligned if the determinant corresponding to them is equal to zero.
Example:
Check if the points A(0,5), B(1,3) and C(2,1) are or not collinear (they are aligned).
The determinant regarding these points is
. In order for them to be collinear, the value of this determinant must equal zero. 
= 10 + 1 – 6 – 5 = 9 – 6 – 5 = 5 – 5 = 0
Therefore, points A, B and C are aligned.
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-utilizando-determinantes.htm
