Cartesian coordinate system on a plane

The Cartesian coordinate system on a plane is a rectangular coordinate system consisting of two mutually perpendicular coordinate lines with selected positive directions of these lines and scales indicated on them. The lines are called coordinate axes, and the point where these lines intersect is the origin. The origin is denoted by the letter O.
Usually the OX axis is a horizontal straight line, the OY axis is a vertical one. The OX axis is called the abscissa axis, the OY axis is the ordinate axis. The coordinate axes divide the plane into 4 quarters, they are indicated by Roman numerals and are counted from the OX ray counterclockwise. The abscissa of point A is the number x, the absolute value of which is equal to the distance from point A to the straight line OY. If point A is located in the right half-plane, then its abscissa x is a positive number, if in the left it is a negative number. If point A belongs to the OY axis, then x=0.
Similarly, the ordinate of point A is a number y, the absolute value of which is equal to the distance from point A to the straight line OX. If point A is located in the upper half-plane, then its ordinate y is a positive number, if in the lower one it is a negative number. If point A belongs to the OX axis, then y=0. The numbers x, y are the coordinates of point A, it is denoted as follows: A(x; y) or A=(x; y).

Coordinates of the middle of the segment

Let the segment AB be given: A=(x₁; y₁), B=(x₂; y₂). We need to find the coordinates (x; y) of its midpoint - point C.
From points A, B and C we drop perpendiculars onto the X-axis, they will intersect the X-axis at points E(x₁; 0), F(x₂; 0), G(x; 0), respectively. According to Thales' theorem, point G is the midpoint of segment EF, that is, EG=GF. Therefore, x-x₁=x₂-x. From here we get the formula for the abscissa of the point G: x=(x₁+x₂)/2. This formula remains valid for all cases of the location of points A and B. Even if AB is parallel to OY, we simply get that the abscissas of points A, B, and C coincide.
To derive the formula for the ordinate of point C, we drop the perpendiculars from points A, B, and C onto the OY axis and similarly find that the ordinate y of point G is y=(y₁+y₂)/2.
Thus, the formulas for the coordinates of the middle of the segment have the form:

x=(x₁+x₂)/2,     y=(y₁+y₂)/2.

The length of the segment, the distance between the points

Let the segment AB be given: A=(x₁; y₁), B=(x₂; y₂). We need to find its length, that is, the distance between points A and B.
From points A and B we drop perpendiculars to the X and Y axes. The line perpendicular to the X axis, passing through point A, intersects the line perpendicular to the Y axis, passing through point B, at point C=(x₁; y₂). Thus, we get the right triangle ABC. We will find its hypotenuse AB using the Pythagorean theorem:

AB² = AC² + BC² <=> AB² = (y₂ - y₁)² + (x₂ - x₁)²

This formula is valid for any case of location of points A and B.
Thus, the square of the distance between two points A(x₁; y₁) and B(x₂; y₂) is determined by the formula:

AB² = (y₂ - y₁)² + (x₂ - x₁)²