When I search online for any list of laws of Cartesian coordinate geometry, I find that it includes quite a few laws such as the law of distance between two points, the law of distance between a point and a line, the law of slope of a straight line and a few additional laws.
But I wanted to find a big list of these laws, after searching with specific words and more deeply I found more laws, but my list is still small, I will include most of what I found in an answer, and I hope my list will be expanded with more laws
It would also be great to have a reference or book with a large number of formulas
The law of distance between points $A:(x_1,y_1)$ and $B:(x_2,y_2)$
$\bar{AB}=\sqrt{(x_1-x_2 )^2+(y_1-y_2 )^2 }$
Coordinates of the midpoint between points $(x_1,y_1)$ and $(x_2,y_2)$
$M:(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$
The coordinates of point $C$ that divides (by internal division) the line segment with the ends of $A(x_1,y_1)$ ,$B(x_2,y_2)$ by $\frac{AC}{BC}=\frac{k}{l}$ are:
$C:(\frac{lx_1+kx_2}{k+l},\frac{ly_1+ky_2}{k+l})$
Coordinates of the center of gravity of a triangle with vertices $A:(x_1,y_1)$, $B:(x_2,y_2)$ and $C:(x_3,y_3)$
$O:(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$
Coordinates of the center of gravity of a group of points $(x_1,y_1),(x_2,y_2),...,(x_n,y_n)$
$O:(\frac{1}{n}\cdot∑_{i=1}^n x_i ,\frac{1}{n}\cdot∑_{i=1}^n y_i )$
The law of slope of a line passing through the points $A:(x_1,y_1)$ and $B:(x_2,y_2)$
$m_{AB}=\frac{y_1-y_2}{x_1-x_2}$
If the slope is zero, the line is parallel to the $xx'$ axis with $y=y_1=y_2=λ$
If the slope is not defined, the line is parallel to the $yy'$ axis with the equation $x=x_1=x_2=λ$
The law of equation of a line with slope m passing through point $(x_0,y_0)$
$y-y_0=m(x-x_0)$
The lines are perpendicular if and only if $m_1×m_2=-1$
The two lines are parallel if and only if $m_1 =m_2$ Miller's Law
The law of slope of a line with equation $ax+by+c=0$ is given as $m=-\frac{a}{b}$
Let $θ$ be the measure of the angle made by the line D with the axis of the separations, which starts from the axis of the separators and calculates counterclockwise, $m_D = tanθ$
$L$ intersects $xx'$ in $(a,0)$, and intersects $yy'$ in $(0,b)$ the equation $L$ is written as:
$\frac{x}{a}+\frac{y}{b}=1$
$L$ is removed from the principle by $P$, and $θ$ is the measure of the angle between P and the axis of the separators, the equation $L$ is written as:
$x\cdot cosθ+y\cdot sinθ-p=0$
The Law of Equation of a Circle Centered at Point $(x_0,y_0)$ and Its Radius $r$
$(x-x_0 )^2+(y-y_0 )^2=r^2$
The law of distance between the point $(x_0,y_0)$ and the line $ax+by+c=0$
$Distance=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$
The equation for the point $(x_0,y_0)$ is $(x-x_0)^2+(y-y_0)^2=0$
The law of distance between point $(a,b)$ and line $y=mx+c$
$Distance=\frac{|ma+c-b|}{\sqrt{1+m^2}}$
The law of distance between parallel lines: $y=mx+b_1,y=mx+b_2$
$Distance=\frac{|b_2-b_1|}{\sqrt{1+m^2}}$
The equation of the tangent to a circle $(x-a)^2+(y-b)^2=R^2$ in
Point $(x_0,y_0)$ on the circumference of the circle:
$(x_0-a)⋅(x-a)+(y_0-b)⋅(y-b)=R^2$
The equation of the Mar plane with three points $(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)$
$a(x– x_0 )+b(y– y_0 )+c(z–z_0 )=0$
The general formula for the equation of a plane is from the form ax+by+cz+d=0 The law of distance between points $A:(x_1,y_1,z_1)$ and $B:(x_2,y_2,z_3)$
$\bar{AB}=\sqrt{(x_1-x_2 )^2+(y_1-y_2 )^2+(z_1-z_2 )^2}$
The coordinates of the midpoint between points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$
$M:(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2})$
The law of distance between the point $(x_0,y_0,z_0)$ and the plane $ax+by+cz+d=0$
$Distance=\frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}$
Intermediate equations of a line:
$x=x_0+at$
$y=y_0+bt$
$z=z_0+ct$
In parabola $y^2=4px$
The equation of the tangent at point $(x_0,y_0)$ that lies on the parabola
$y\cdot y_0=2p(x+x_0)$
Parabola guide $x=-p$
Parabola focus $F:(p,0)$
Ellipse equation
$\frac{(x-x_0)^2}{a^2} +\frac{(y-y_0)^2}{b^2}=1$
coordinates $O:(x_0,y_0)$ Focal distance from center of ellipsis
c=\sqrt{|a^2-b^2|)}
The length of the large diameter is $2a$ in case of $a>b$ is $2b$ and is equal to in case of $b>a$
Hyperbola equation
$±\frac{(x-x_0 )^2}{a^2}∓\frac{(y-y_0 )^2}{b^2}=1$
coordinates $O:(x_0,y_0)$
Focal distance from the center of the ellipsis
$c=\sqrt{a^2+b^2}$
General equation of a conical section
$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$
The conic segments described in this equation can be classified using the characteristic
$B^2-4AC$
If the cone is not decaying, then The equation represents an ellipse if $B^2-4AC<0$ and if $A=C$ and $B=0$ it is a special case of an ellipse and the equation represents a circle and if $B^2-4AC=0$ the equation represents a parabola and if $B^2-4AC>0$ the equation represents a hyperbola If we also have $A+C=0$, then the equation represents an isosceles hyperbola
In polar coordinates, the equation of a circle with center $(a_0,θ_0)$ and radius $r$ is:
$a^2+(a_0)^2-2a\cdot a_0\cdot cos(θ-θ_0 )=r^2$
The Cartesian equation of a three-point circle $(x_1,y_1,(x_2,y_2),(x_3,y_3)$
$\frac{(x-x_1 )(x-x_2 )+(y-y_1 )(y-y_2 )}{(y-y_1 )(x-x_2 )-(y-y_2 )(x-x_1 ) }=\frac{(x_3-x_1 )(x_3-x_2 )+(y_3-y_1 )(y_3-y_2 )}{(y_3-y_1 )(x_3-x_2 )-(y_3-y_2 )(x_3-x_1) )}$
The bisector equations of the lines $a_1 x+b_1 y+c_1=0$ and $a_2 x+b_2 y+c_2=0$ are:
$\frac{a_1 x+b_1 y+c_1}{\sqrt{(a_1 )^2+(b_1 )^2}}=±\frac{a_2 x+b_2 y+c_2}{\sqrt{(a_2 )^2+(b_2 )^2}}$
The equation of a line segment limited by points (x_1,y_1) and (x_2,y_2) is:
$\sqrt{(x-x_1 )^2+(y-y_1 )^2}+\sqrt{(x-x_2 )^2+(y-y_2 )^2}=\sqrt{(x_1-x_2 )^2+(y_1-y_2 )^2}$
The equation of the line from which the line segment is excluded is limited by the points $(x_1,y_1)$ and $(x_2,y_2)$ is:
$|\sqrt{(x-x_1)^2+(y-y_1)^2}-\sqrt{(x-x_2 )^2+(y-y_2 )^2}|=\sqrt{(x_1-x_2)^2+(y_1-y_2 )^2}$